# Equational theories of idempotent semifields
**Authors**: G. Metcalfe and S. Santschi
> george.metcalfe@unibe.chsimon.santschi@unibe.ch
\newnumbered
assertionAssertion \newnumbered conjectureConjecture \newnumbered definitionDefinition \newnumbered hypothesisHypothesis \newnumbered remarkRemark \newnumbered noteNote \newnumbered observationObservation \newnumbered problemProblem \newnumbered questionQuestion \newnumbered algorithmAlgorithm \newnumbered exampleExample \newunnumbered notationNotation \simpleequations \classno 12K10, 16Y60, 03C05, 06F15 \extraline Supported by Swiss National Science Foundation (SNSF) grant No. 200021_215157.
Abstract
This paper provides answers to several open problems about equational theories of idempotent semifields. In particular, it is proved that (i) no equational theory of a non-trivial class of idempotent semifields has a finite basis; (ii) there are continuum-many equational theories of classes of idempotent semifields; and (iii) the equational theory of the class of idempotent semifields is co-NP-complete. This last result is also used to determine the complexity of deciding the existence of a right order on a free group or free monoid satisfying finitely many given inequalities.
1 Introduction
An idempotent semiring is an algebraic structure $\langle{S,\vee,·,{\rm e}}\rangle$ satisfying
1. $\langle{S,·,{\rm e}}\rangle$ is a monoid;
1. $\langle{S,\vee}\rangle$ is a semilattice (i.e., an idempotent commutative semigroup); and
1. $a(b\vee c)d=abd\vee acd$ for all $a,b,c,d∈ S$ .
If $\langle{S,·,{\rm e}}\rangle$ is the monoid reduct of a group, then $\langle{S,\vee,·,{\rm e}}\rangle$ is called an idempotent semifield. Such structures arise naturally in many areas of mathematics, including idempotent analysis, formal language theory, tropical geometry, and mathematical logic; for details and further references, see [6, 7, 8]. Alternative definitions of an idempotent semiring (also known as a dioid or an ai-semiring) may be found in the literature that differ with respect to the presence and/or role of constant symbols in the signature. In particular, an idempotent semiring is sometimes defined without ${\rm e}$ , where $\langle{S,·}\rangle$ is required to be a semigroup, and sometimes with both ${\rm e}$ and a further constant symbol $0 0$ , where $0 0$ is interpreted as the neutral element of $\vee$ . In the latter case, the definition of an idempotent semifield is changed so that $\langle{S{\setminus}\{0\},·,{\rm e}}\rangle$ is the monoid reduct of a group. As explained in Section 5, however, our results extend also to these settings.
Any idempotent semifield $\langle{S,\vee,·,{\rm e}}\rangle$ expanded with the group inverse operation -1 and lattice meet operation $\wedge$ defined by setting $a\wedge b:={({a}^{-1}\vee{b}^{-1})}^{-1}$ is a lattice-ordered group ( $\ell$ -group, for short): an algebraic structure $\langle{L,\wedge,\vee,·,{}^{-1},{\rm e}}\rangle$ such that $\langle{L,·,{}^{-1},{\rm e}}\rangle$ is a group; $\langle{L,\wedge,\vee}\rangle$ is a lattice with an order defined by $x≤ y\>:\Longleftrightarrow\>x\vee y=y$ ; and multiplication is order-preserving, i.e., $a≤ b\>\Longrightarrow\>cad≤ cbd$ , for all $a,b,c,d∈ L$ . Indeed, idempotent semifields are precisely the semiring reducts of $\ell$ -groups.
In this paper, we provide answers to several open problems about equational theories of idempotent semifields and related structures. Although these problems have been solved for $\ell$ -groups, restricting to fewer operations requires the development of new proof methods and yields notably different results.
In order to present these results, let us first recall some basic terminology. A signature ${\mathcal{L}}$ is a set of operation symbols with finite arities, and an ${\mathcal{L}}$ -algebra $\mathbf{A}$ consists of a non-empty set $A$ equipped with an $n$ -ary function on $A$ for each operation symbol of ${\mathcal{L}}$ of arity $n$ . An ${\mathcal{L}}$ -term is built inductively using the operation symbols of ${\mathcal{L}}$ and a countably infinite set of variables, and the ${\mathcal{L}}$ -term algebra $\mathbf{Tm}({\mathcal{L}})$ consists of the set of ${\mathcal{L}}$ -terms equipped with the term-building operation symbols of ${\mathcal{L}}$ . An ${\mathcal{L}}$ -equation is an ordered pair of ${\mathcal{L}}$ -terms $s,t$ , written $s≈ t$ , and is satisfied by an ${\mathcal{L}}$ -algebra $\mathbf{A}$ , written $\mathbf{A}\models s≈ t$ , if $\varphi(s)=\varphi(t)$ , for any homomorphism $\varphi\colon\mathbf{Tm}({\mathcal{L}})→\mathbf{A}$ . Given any class of ${\mathcal{L}}$ -algebras $\mathsf{K}$ and set of ${\mathcal{L}}$ -equations $\mathrm{\Sigma}$ , we denote by $\mathsf{K}\models\mathrm{\Sigma}$ that $\mathbf{A}\models s≈ t$ for all $\mathbf{A}∈\mathsf{K}$ and $s≈ t∈\mathrm{\Sigma}$ . The equational theory ${\rm Eq}(\mathsf{K})$ of a class of ${\mathcal{L}}$ -algebras $\mathsf{K}$ is the set of all ${\mathcal{L}}$ -equations $s≈ t$ such that $\mathsf{K}\models s≈ t$ .
In Section 2, we provide a complete answer to the finite basis problem for idempotent semifields. Let $\mathsf{K}$ be any class of ${\mathcal{L}}$ -algebras, and call it non-trivial if at least one of its members is non-trivial, i.e., has more than one element. A basis for the equational theory of $\mathsf{K}$ is a set of equations $\mathrm{\Sigma}⊂eq{\rm Eq}(\mathsf{K})$ such that every equation in ${\rm Eq}(\mathsf{K})$ is a logical consequence of $\mathrm{\Sigma}$ , that is, if $\mathbf{A}\models\mathrm{\Sigma}$ for some ${\mathcal{L}}$ -algebra $\mathbf{A}$ , then $\mathbf{A}\models{\rm Eq}(\mathsf{K})$ . If ${\rm Eq}(\mathsf{K})$ has a finite basis, then $\mathsf{K}$ is said to be finitely based. Notably, the equational theory of the $\ell$ -group $\langle{{\mathbb{Z}},\min,\max,+,-,0}\rangle$ is finitely based, but this is not the case for the semifield $\langle{{\mathbb{Z}},\max,+,0}\rangle$ or any other totally ordered semifield [1, Theorem 48]. Indeed, although countably infinitely many equational theories of $\ell$ -groups have a finite basis (see, e.g., [2, Section 7.2]), we prove here that:
{t:basis*}
There is no non-trivial class of idempotent semifields that is finitely based.
In Section 3, we determine the number of equational theories of classes of idempotent semifields. Using a technique of ‘inverse elimination’ to translate between equations in the different signatures, we obtain a one-to-one correspondence between a family of equational theories of $\ell$ -groups that is known to be uncountable and equational theories of certain classes of idempotent semifields, thereby proving:
{t:continuum*}
There are continuum-many equational theories of classes of idempotent semifields.
In Section 4, we establish the complexity of deciding equations in the class of idempotent semifields. The equational theory of the class of $\ell$ -groups is known to be co-NP-complete [5, Theorem 8.3] and we prove here that this is also the case for the restricted signature, that is:
{t:conp*}
The equational theory of the class of idempotent semifields is co-NP-complete.
We also use this result to show that the problem of deciding if there exists a right order on the free $n$ -generated group ( $n≥ 2$ ) whose positive cone contains a given finite set of elements is NP-complete and that the same is true for the problem of deciding if there exists a right order on the free countably infinitely generated monoid that satisfies a given finite set of inequalities.
Finally, in Section 5, we extend the results of the paper to related structures considered in the literature: expansions of idempotent semifields with the meet operation, ${\rm e}$ -free reducts of idempotent semifields, and idempotent semifields extended with a neutral element $0 0$ for the join operation.
2 The finite basis problem
Let ${\mathcal{L}}_{m}$ and ${\mathcal{L}}_{s}$ be the signatures of monoids and idempotent semirings, respectively. Following [11], let the flat extension of an ${\mathcal{L}}_{m}$ -algebra $\mathbf{M}=\langle{M,·,{\rm e}}\rangle$ be the ${\mathcal{L}}_{s}$ -algebra $\flat(\mathbf{M})=\langle{M\cup\{→p\},\vee,\star,{\rm e}}\rangle$ , where $→p\not∈ M$ and for all $a,b∈ M\cup\{→p\}$ ,
$$
a\star b:=\begin{cases}a\cdot b&\text{if }a,b\in M\\
\top&\text{otherwise}\end{cases}\qquad\text{and}\qquad a\vee b:=\begin{cases}a%
&\text{if }a=b\\
\top&\text{otherwise.}\end{cases}
$$
It is easily confirmed that if $\mathbf{M}$ is a monoid, then $\langle{M\cup\{→p\},\star,{\rm e}}\rangle$ is a monoid, $\langle{M\cup\{→p\},\vee}\rangle$ is a semilattice of height one, and $→p$ is an absorbing element for both the binary operations of $\flat(\mathbf{M})$ . The following result provides a necessary and sufficient condition for $\flat(\mathbf{M})$ to be an idempotent semiring.
**Lemma 2.1 (cf.[12, Lemma 2.2])**
*Let $\mathbf{M}$ be any monoid. Then $\flat(\mathbf{M})$ is an idempotent semiring if and only if $\mathbf{M}$ is cancellative, i.e., $cad=cbd\>\Longrightarrow\>a=b$ , for all $a,b,c,d∈ M$ .*
**Proof 2.2**
*Suppose first that $\flat(\mathbf{M})$ is an idempotent semiring and $cad=cbd$ for some $a,b,c,d∈ M$ . Then $c(a\vee b)d=cad\vee cbd=cad≠→p$ , so $a\vee b≠→p$ and $a=b$ . For the converse, suppose that $\mathbf{M}$ is cancellative and consider any $a,b,c,d∈ M$ . If $b=c$ , then, clearly, $a(b\vee c)d=abd\vee acd$ . Otherwise, $b≠c$ and, by cancellativity, $abd≠acd$ , so $a(b\vee c)d=a→p d=→p=abd\vee acd$ .*
Consider now the monoid reduct $\mathbf{Z}=\langle{{\mathbb{Z}},+,0}\rangle$ of the additive integer group, and monoid reducts $\mathbf{Z}_{n}=\langle{Z_{n},·,{\rm e}}\rangle$ of the cyclic groups of order $n∈{\mathbb{N}}^{>0}$ . Since these monoids are cancellative, their flat extensions, $\flat(\mathbf{Z})$ and $\flat(\mathbf{Z}_{n})$ ( $n∈{\mathbb{N}}^{>0}$ ), are idempotent semirings with a flat semilattice structure, as depicted in Figure 1. We will prove first that the equational theory of any finitely based class of idempotent semirings $\mathsf{K}$ is satisfied by $\flat(\mathbf{Z})$ if and only if it is satisfied by $\flat(\mathbf{Z}_{p})$ for every prime $p$ greater than some suitably large $n∈{\mathbb{N}}$ (Corollary 2.11). We will then show that the equational theory of any non-trivial class of idempotent semifields is satisfied by $\flat(\mathbf{Z})$ , but not by $\flat(\mathbf{Z}_{n})$ for any $n∈{\mathbb{N}}^{>0}$ , thereby establishing that the class is not finitely based (Theorem A).
$→p$ $-2$ $-1$ $0 0$ $1$ $2$ $...$ %**** MetSan˙Dec6.tex Line 225 **** $...$
$→p$ $...$ ${\rm e}$ $a$ $a^{2}$ $a^{n-1}$
Figure 1: Hasse diagrams of $\flat(\mathbf{Z})$ and $\flat(\mathbf{Z}_{n})$ .
For a signature ${\mathcal{L}}$ containing the binary operation symbol $\vee$ , we call an ${\mathcal{L}}$ -equation of the form $s\vee t≈ t$ , often written $s≤ t$ , an ${\mathcal{L}}$ -inequation. We call an ${\mathcal{L}}_{s}$ -inequation simple if it is of the form $s≤ t_{1}\vee·s\vee t_{n}$ , where $s,t_{1},...,t_{n}$ are ${\mathcal{L}}_{m}$ -terms, and call this simple ${\mathcal{L}}_{s}$ -inequation left-regular if each variable occurring in $s$ occurs in at least one of $t_{1},...,t_{n}$ .
**Remark 2.3**
*Clearly, an idempotent semiring satisfies an ${\mathcal{L}}_{s}$ -equation $s≈ t$ if and only it satisfies the ${\mathcal{L}}_{s}$ -inequations $s≤ t$ and $t≤ s$ . Hence, using the distributivity of multiplication over binary joins and the fact that $a\vee b≤ c\iff a≤ c\text{ and }b≤ c$ for all elements $a,b,c$ of an idempotent semiring, there exists an algorithm that produces for every ${\mathcal{L}}_{s}$ -equation $\varepsilon$ , a finite set $\Sigma$ of simple ${\mathcal{L}}_{s}$ -inequations such that an arbitrary idempotent semiring satisfies $\varepsilon$ if and only if it satisfies $\mathrm{\Sigma}$ .*
**Remark 2.4**
*If an idempotent semiring satisfies a simple ${\mathcal{L}}_{s}$ -inequation that is not left-regular, then — by substituting all variables, except for one that occurs on the left and not the right, with ${\rm e}$ — it must also satisfy $x^{n}≤{\rm e}$ for some $n∈{\mathbb{N}}^{>0}$ . Hence, reasoning contrapositively, if a simple ${\mathcal{L}}_{s}$ -inequation is satisfied by an idempotent semiring containing an element greater than ${\rm e}$ , it must be left-regular.*
For a signature ${\mathcal{L}}$ , an ${\mathcal{L}}$ -quasiequation is an ordered pair consisting of a finite set of ${\mathcal{L}}$ -equations $\mathrm{\Sigma}$ and an ${\mathcal{L}}$ -equation $s≈ t$ , written $\mathrm{\Sigma}\Rightarrow s≈ t$ , and is satisfied by an ${\mathcal{L}}$ -algebra $\mathbf{A}$ if for any homomorphism $\varphi\colon\mathbf{Tm}({\mathcal{L}})→\mathbf{A}$ , whenever $\varphi(s^{\prime})=\varphi(t^{\prime})$ for all $s^{\prime}≈ t^{\prime}∈\mathrm{\Sigma}$ , also $\varphi(s)=\varphi(t)$ . Given any simple ${\mathcal{L}}_{s}$ -inequation $\varepsilon=(s≤ t_{1}\vee·s\vee t_{n})$ , we define a corresponding ${\mathcal{L}}_{m}$ -quasiequation
$$
Q(\varepsilon):=\{t_{1}\approx t_{2},\dots,t_{1}\approx t_{n}\}\Rightarrow t_{%
1}\approx s.
$$
**Lemma 2.5**
*Let $\mathbf{M}$ be any monoid. Then for any left-regular simple ${\mathcal{L}}_{s}$ -inequation $\varepsilon$ ,
$$
\flat(\mathbf{M})\models\varepsilon\iff\mathbf{M}\models Q(\varepsilon).
$$*
**Proof 2.6**
*Let $\varepsilon=(s≤ t_{1}\vee·s\vee t_{n})$ be any left-regular simple ${\mathcal{L}}_{s}$ -inequation. For the left-to-right direction, suppose contrapositively that $\mathbf{M}\not\models Q(\varepsilon)$ . Then $\varphi(t_{1})=·s=\varphi(t_{n})$ and $\varphi(t_{1})≠\varphi(s)$ for some homomorphism $\varphi\colon\mathbf{Tm}({\mathcal{L}}_{m})→\mathbf{M}$ . Let $\hat{\varphi}\colon\mathbf{Tm}({\mathcal{L}}_{s})→\flat(\mathbf{M})$ be the homomorphism defined by setting $\hat{\varphi}(x):=\varphi(x)$ for every variable $x$ . Clearly, $\hat{\varphi}(u)=\varphi(u)$ for each ${\mathcal{L}}_{m}$ -term $u$ and therefore $\hat{\varphi}(t_{1}\vee·s\vee t_{n})=\hat{\varphi}(t_{1})\vee·s\vee%
\hat{\varphi}(t_{n})=\varphi(t_{1})\vee·s\vee\varphi(t_{n})=\varphi(t_{1})%
≠\varphi(s)=\hat{\varphi}(s)$ . But $\varphi(t_{1}),\varphi(s)∈ M$ , so $\hat{\varphi}(t_{1}\vee·s\vee t_{n})\not≤\hat{\varphi}(s)$ . Hence $\flat(\mathbf{M})\not\models\varepsilon$ . For the converse direction, suppose contrapositively that $\flat(\mathbf{M})\not\models\varepsilon$ . Then $\psi(s)\nleq\psi(t_{1}\vee·s\vee t_{n})$ for some homomorphism $\psi\colon\mathbf{Tm}({\mathcal{L}}_{s})→\flat(\mathbf{M})$ . It follows from the definition of the order of $\flat(\mathbf{M})$ that $\psi(t_{1}\vee·s\vee t_{n})≠→p$ and $\psi(t_{1})=·s=\psi(t_{n})≠→p$ . Hence, since $\varepsilon$ is left-regular, $\psi(x)∈ M$ for every variable $x$ occurring in $\varepsilon$ , and there exists a homomorphism $\hat{\psi}\colon\mathbf{Tm}({\mathcal{L}}_{m})→\mathbf{M}$ satisfying $\hat{\psi}(x)=\psi(x)$ for all such $x$ . Clearly, $\hat{\psi}(t_{1})=·s=\hat{\psi}(t_{n})$ and $\hat{\psi}(t_{1})≠\hat{\psi}(s)$ . So $\mathbf{M}\not\models Q(\varepsilon)$ .*
**Remark 2.7**
*The right-to-left direction of Lemma 2.5 does not hold for all simple ${\mathcal{L}}_{s}$ -inequations that are not left-regular; e.g., $\mathbf{Z}_{2}\models\emptyset\Rightarrow{\rm e}≈ x^{2}$ , but $\flat(\mathbf{Z}_{2})\not\models x^{2}≤{\rm e}$ .*
**Lemma 2.8**
*Let $\mathrm{\Delta}$ be any finite set of ${\mathcal{L}}_{m}$ -quasiequations. Then $\mathbf{Z}\models\mathrm{\Delta}$ if and only if there exists an $n∈{\mathbb{N}}$ such that $\mathbf{Z}_{p}\models\mathrm{\Delta}$ for every prime $p>n$ .*
**Proof 2.9**
*For the right-to-left direction, suppose that there exists an $n∈{\mathbb{N}}$ such that $\mathbf{Z}_{p}\models\mathrm{\Delta}$ for every $p∈ P_{n}:=\{p∈{\mathbb{N}}\mid\text{$p>n$ and $p$ is prime}\}$ . Since $\mathrm{\Delta}$ is a set of ${\mathcal{L}}_{m}$ -quasiequations, $\prod_{p∈ P_{n}}\mathbf{Z}_{p}\models\mathrm{\Delta}$ . The projection maps $\pi_{p}\colon\mathbf{Z}→\mathbf{Z}_{p}$ $(p∈ P_{n})$ induce an embedding of $\mathbf{Z}$ into $\prod_{p∈ P_{n}}\mathbf{Z}_{p}$ , so also $\mathbf{Z}\models\mathrm{\Delta}$ . For the converse direction, suppose that $\mathbf{Z}\models\mathrm{\Delta}$ . Let $T$ be a set of first-order sentences axiomatizing the class of Abelian groups and let $\alpha$ be the conjunction of the ${\mathcal{L}}_{m}$ -quasiequations in $\mathrm{\Delta}$ . Since an ${\mathcal{L}}_{m}$ -quasiequation is satisfied by all torsion-free Abelian groups if and only if it is satisfied by $\mathbf{Z}$ , it follows that $T\cup\{\{x^{k}≈{\rm e}\}\Rightarrow x≈{\rm e}\mid k∈{\mathbb{N}}%
\}\models\alpha$ . By compactness, there exists an $n∈{\mathbb{N}}$ such that $T\cup\{\{x^{k}≈{\rm e}\}\Rightarrow x≈{\rm e}\mid k∈{\mathbb{N}}%
\text{ and }k≤ n\}\models\alpha$ . Hence $\mathbf{Z}_{p}\models\mathrm{\Delta}$ for every prime $p>n$ .*
**Proposition 2.10**
*Let $\mathrm{\Sigma}$ be any finite set of left-regular simple ${\mathcal{L}}_{s}$ -inequations. Then $\flat(\mathbf{Z})\models\mathrm{\Sigma}$ if and only if there exists an $n∈{\mathbb{N}}$ such that $\flat(\mathbf{Z}_{p})\models\mathrm{\Sigma}$ for every prime $p>n$ .*
{proof*}
Let $\mathrm{\Delta}:=\{Q(\varepsilon)\mid\varepsilon∈\mathrm{\Sigma}\}$ . Then
| | $\displaystyle\flat(\mathbf{Z})\models\mathrm{\Sigma}$ | $\displaystyle\iff\mathbf{Z}\models\mathrm{\Delta}$ | (Lemma 2.5) | |
| --- | --- | --- | --- | --- |
**Corollary 2.11**
*Let $\mathsf{K}$ be any finitely based class of idempotent semirings. Then $\flat(\mathbf{Z})\models{\rm Eq}(\mathsf{K})$ if and only if there exists an $n∈{\mathbb{N}}$ such that $\flat(\mathbf{Z}_{p})\models{\rm Eq}(\mathsf{K})$ for every prime $p>n$ .*
**Proof 2.12**
*By assumption and Remark 2.3, there exists a finite basis $\mathrm{\Sigma}$ for ${\rm Eq}(\mathsf{K})$ that consists of simple ${\mathcal{L}}_{s}$ -inequations. Moreover, by Remark 2.4, every simple ${\mathcal{L}}_{s}$ -inequation satisfied by $\flat(\mathbf{M})$ for some monoid $\mathbf{M}$ is left-regular. Hence $\flat(\mathbf{Z})\models\mathrm{\Sigma}$ if and only there exists an $n∈{\mathbb{N}}$ such that $\flat(\mathbf{Z}_{p})\models\mathrm{\Sigma}$ for every prime $p>n$ , by Proposition 2.10. The claim therefore follows directly using the fact that $\mathrm{\Sigma}$ is a basis for ${\rm Eq}(\mathsf{K})$ .*
**Theorem A**
*There is no non-trivial class of idempotent semifields that is finitely based.*
**Proof 2.13**
*Let $\mathsf{K}$ be any non-trivial class of idempotent semifields. Consider first any $n∈{\mathbb{N}}^{>0}$ . Since the ${\mathcal{L}}_{s}$ -inequation $x≤{\rm e}\vee x^{n}$ is satisfied by all $\ell$ -groups, it belongs to ${\rm Eq}(\mathsf{K})$ . On the other hand, $\mathbf{Z}_{n}\not\models\{x^{n}≈{\rm e}\}\Rightarrow x≈{\rm e}$ yields $\flat(\mathbf{Z}_{n})\not\models x≤{\rm e}\vee x^{n}$ , by Lemma 2.5, so $\flat(\mathbf{Z}_{n})\not\models{\rm Eq}(\mathsf{K})$ . Hence, by Corollary 2.11, to show that $\mathsf{K}$ is not finitely based, it suffices to prove that $\flat(\mathbf{Z})\models{\rm Eq}(\mathsf{K})$ . Moreover, by Remark 2.3, it suffices to show that $\flat(\mathbf{Z})$ satisfies every simple ${\mathcal{L}}_{s}$ -inequation in ${\rm Eq}(\mathsf{K})$ , recalling that, by Remark 2.4, since $\mathsf{K}$ is non-trivial, these simple ${\mathcal{L}}_{s}$ -inequations are all left-regular. Let $\varepsilon=(s≤ t_{1}\vee·s\vee t_{n})$ be any left-regular simple ${\mathcal{L}}_{s}$ -inequation and suppose contrapositively that $\flat(\mathbf{Z})\not\models\varepsilon$ . Then $\mathbf{Z}\not\models Q(\varepsilon)$ , by Lemma 2.5, i.e., there exists a homomorphism $\varphi\colon\mathbf{Tm}({\mathcal{L}}_{m})→\mathbf{Z}$ such that $\varphi(t_{1})=·s=\varphi(t_{n})$ and $\varphi(t_{1})≠\varphi(s)$ . Moreover, we can assume that $\varphi(t_{1})<\varphi(s)$ in the standard order on ${\mathbb{Z}}$ , and let $\hat{\varphi}\colon\mathbf{Tm}({\mathcal{L}}_{s})→\langle{{\mathbb{Z}},\max,%
+,0}\rangle$ be the homomorphism defined by setting $\hat{\varphi}(x):=\varphi(x)$ for each variable $x$ . Then $\hat{\varphi}(s)=\varphi(s)>\varphi(t_{1})=\hat{\varphi}(t_{1})=\max\{\hat{%
\varphi}(t_{1}),...,\hat{\varphi}(t_{n})\}=\hat{\varphi}(t_{1}\vee·s\vee
t%
_{n})$ . So $\langle{{\mathbb{Z}},\max,+,0}\rangle\not\models\varepsilon$ . Now consider any non-trivial $\mathbf{A}∈\mathsf{K}$ and $a∈ A$ with $a>{\rm e}$ . The homomorphism induced by mapping $1$ to $a$ embeds $\langle{{\mathbb{Z}},\max,+,0}\rangle$ into $\mathbf{A}$ , so also $\mathbf{A}\not\models\varepsilon$ and $\varepsilon\not∈{\rm Eq}(\mathsf{K})$ .*
Let us remark finally that the approach followed in this section to establish Theorem A extends to a broader class of idempotent semirings. More precisely, there is no finitely based class of idempotent semirings $\mathsf{K}$ such that $\flat(\mathbf{Z})\models{\rm Eq}(\mathsf{K})$ and $\mathsf{K}\models x≤{\rm e}\vee x^{n}$ for every $n∈{\mathbb{N}}^{>0}$ . In particular, the class of totally ordered idempotent semirings is not finitely based, which follows also from [1, Theorem 48].
3 The number of equational theories
For any countable signature ${\mathcal{L}}$ and class of ${\mathcal{L}}$ -algebras $\mathsf{K}$ , there can be at most continuum-many equational theories of subclasses of $\mathsf{K}$ . In particular, although there are just two equational theories of classes of commutative idempotent semifields — ${\rm Eq}(\langle{{\mathbb{Z}},\max,+,0}\rangle)$ and the set of all ${\mathcal{L}}_{s}$ -equations — the maximum number is attained in the setting of commutative idempotent semirings.
**Proposition 3.1**
*There are continuum-many equational theories of classes of commutative idempotent semirings.*
**Proof 3.2**
*For any set of primes $P$ , let $\mathrm{\Sigma}_{P}$ denote the equational theory of the class of commutative idempotent semirings satisfying $x≤{\rm e}\vee x^{p}$ for all $p∈ P$ . We show that $\mathrm{\Sigma}_{P}≠\mathrm{\Sigma}_{Q}$ for any two distinct sets of primes $P$ and $Q$ , and hence that there are continuum-many such equational theories. Consider, without loss of generality, $p∈ P{\setminus}Q$ . Then $x≤{\rm e}\vee x^{p}∈\mathrm{\Sigma}_{P}$ . But also, since $\mathbf{Z}_{p}\not\models\{{\rm e}≈ x^{p}\}\Rightarrow{\rm e}≈ x$ and $\mathbf{Z}_{p}\models\{{\rm e}≈ x^{q}\}\Rightarrow{\rm e}≈ x$ for all $q∈ Q$ , it follows from Lemma 2.5 that $\flat(\mathbf{Z}_{p})\not\models x≤{\rm e}\vee x^{p}$ and $\flat(\mathbf{Z}_{p})\models x≤{\rm e}\vee x^{q}$ for all $q∈ Q$ . So $x≤{\rm e}\vee x^{p}\not∈\mathrm{\Sigma}_{Q}$ .*
To prove that the maximum number of equational theories is attained also in the setting of idempotent semifields (Theorem B), we make use of a corresponding result for a certain class of $\ell$ -groups. Note first that, unlike idempotent semifields, $\ell$ -groups form a variety: a class of algebras of the same signature that is closed under taking homomorphic images, subalgebras, and direct products — or, equivalently, by Birkhoff’s theorem, an equational class. Equational theories of classes of $\ell$ -groups are therefore in one-to-one correspondence with varieties of $\ell$ -groups. In particular, equational theories of classes of totally ordered groups correspond to varieties of $\ell$ -groups that are representable, that is, subalgebras of direct products of totally ordered groups. For further details and references, we refer to [2].
Let ${\mathcal{L}}_{g}$ and ${\mathcal{L}}_{\ell}$ be the signatures of groups and $\ell$ -groups, respectively. A crucial role in our proof of Theorem B will be played by the following result, recalling that a variety $\mathsf{V}_{1}$ is defined relative to a variety $\mathsf{V}_{2}$ by a set of equations $\mathrm{\Sigma}$ if $\mathsf{V}_{1}$ consists of all the members of $\mathsf{V}_{2}$ that satisfy $\mathrm{\Sigma}$ :
**Theorem 3.3 ([13, Theorem 1])**
*There are continuum-many varieties defined relative to the variety of representable $\ell$ -groups by a set of ${\mathcal{L}}_{g}$ -equations.*
To prove that there are continuum-many equational theories of classes of idempotent semifields, it therefore suffices, by Theorem 3.3, to show that any two varieties defined relative to the variety of representable $\ell$ -groups by sets of ${\mathcal{L}}_{g}$ -equations can be distinguished by an ${\mathcal{L}}_{s}$ -equation. As we show below, such equations can be obtained by ‘eliminating inverses’ from ${\mathcal{L}}_{\ell}$ -inequations.
Let us say that a variety $\mathsf{V}$ of $\ell$ -groups has the product-splitting property if for any ${\mathcal{L}}_{\ell}$ -terms $s,t,u$ and any variable $y$ that does not occur in $s,t,u$ ,
$$
\mathsf{V}\models{\rm e}\leq u\vee st\iff\mathsf{V}\models{\rm e}\leq u\vee sy%
\vee{y}^{-1}t.
$$
**Lemma 3.4**
*Let $\mathsf{V}$ be any variety that is defined relative to the variety of representable $\ell$ -groups by a set of ${\mathcal{L}}_{g}$ -equations. Then $\mathsf{V}$ has the product-splitting property.*
**Proof 3.5**
*Consider any ${\mathcal{L}}_{\ell}$ -terms $s,t,u$ and any variable $y$ that does not occur in $s,t,u$ . Suppose first that $\mathsf{V}\models{\rm e}≤ u\vee st$ . Every $\ell$ -group satisfies the ${\mathcal{L}}_{s}$ -quasiequation $\{{\rm e}≤ x\vee yz\}\Rightarrow{\rm e}≤ x\vee y\vee z$ (cf. [5, Lemma 3.3]), so also $\mathsf{V}\models{\rm e}≤ u\vee sy\vee{y}^{-1}t$ . Now suppose that $\mathsf{V}\not\models{\rm e}≤ u\vee st$ . Since $\mathsf{V}$ is a variety of representable $\ell$ -groups, there exists a totally ordered group $\mathbf{L}∈\mathsf{V}$ and homomorphism $\varphi\colon\mathbf{Tm}({\mathcal{L}}_{\ell})→\mathbf{L}$ satisfying ${\rm e}>\varphi(u\vee st)=\varphi(u)\vee\varphi(s)\varphi(t)$ and therefore also ${\rm e}>\varphi(u)$ and ${\varphi(s)}^{-1}>\varphi(t)$ . Let $\mathbf{M}$ be the totally ordered group consisting of the direct product of the group reducts of $\mathbf{L}$ and $\mathbf{Q}=\langle{{\mathbb{Q}},\min,\max,+,-,0}\rangle$ equipped with the lexicographic order on $L×{\mathbb{Q}}$ . Clearly, $\psi\colon\mathbf{L}→\mathbf{M};\>a\mapsto\langle{a,0}\rangle$ is an embedding of $\mathbf{L}$ into $\mathbf{M}$ . Note that $\mathbf{Q}∈\mathsf{V}$ , since the variety of Abelian $\ell$ -groups is the unique atom in the subvariety lattice of the variety of $\ell$ -groups (see [2]). Moreover, since the group reduct of $\mathbf{M}$ is the direct product of the group reducts of $\mathbf{L}$ and $\mathbf{Q}$ , any ${\mathcal{L}}_{g}$ -equation satisfied by $\mathbf{L}$ and $\mathbf{Q}$ — in particular, those defining $\mathsf{V}$ relative to the variety of representable $\ell$ -groups — is satisfied by $\mathbf{M}$ . So $\mathbf{M}∈\mathsf{V}$ . Now let $\hat{\varphi}\colon\mathbf{Tm}({\mathcal{L}}_{\ell})→\mathbf{M}$ be the homomorphism defined by setting $\hat{\varphi}(y):=\langle{\varphi(t),1}\rangle$ and $\hat{\varphi}(x)=\psi\varphi(x)=\langle{\varphi(x),0}\rangle$ for each variable $x≠y$ . Since $y$ does not occur in $s,t,u$ , clearly $\hat{\varphi}(u)=\langle{\varphi(u),0}\rangle$ , ${\hat{\varphi}(s)}^{-1}=\langle{{\varphi(s)}^{-1},0}\rangle$ , and $\hat{\varphi}(t)=\langle{\varphi(t),0}\rangle$ . So $\hat{\varphi}({\rm e})=\langle{{\rm e},0}\rangle>\hat{\varphi}(u)$ and ${\hat{\varphi}(s)}^{-1}>\hat{\varphi}(y)>\hat{\varphi}(t)$ , yielding $\hat{\varphi}({\rm e})=\langle{{\rm e},0}\rangle>\hat{\varphi}(s)\hat{\varphi}%
(y)=\hat{\varphi}(sy)$ and $\hat{\varphi}({\rm e})=\langle{{\rm e},0}\rangle>{\hat{\varphi}(y)}^{-1}\hat{%
\varphi}(t)=\hat{\varphi}({y}^{-1}t)$ , and then, combining these inequalities, $\hat{\varphi}({\rm e})=\langle{{\rm e},0}\rangle>\hat{\varphi}(u\vee sy\vee{y}%
^{-1}t)$ . Hence $\mathsf{V}\not\models{\rm e}≤ u\vee sy\vee{y}^{-1}t$ .*
**Remark 3.6**
*The proof of Lemma 3.4 establishes that every variety $\mathsf{V}$ defined relative to the variety of representable $\ell$ -groups by a set of ${\mathcal{L}}_{g}$ -equations is densifiable, that is, every totally ordered member of $\mathsf{V}$ embeds into a dense totally ordered member of $\mathsf{V}$ . Clearly, every densifiable variety of representable $\ell$ -groups has the product-splitting property; indeed, densifiability is equivalent to a slightly stronger version of this property in the broader setting of semilinear residuated lattices (see [15] for details).*
Observe next that if $\mathsf{V}$ is a variety of $\ell$ -groups that has the product-splitting property, then for any ${\mathcal{L}}_{\ell}$ -terms $r,s,t,u,v$ and variable $y$ that does not occur in $r,s,t,u,v$ ,
| | $\displaystyle\mathsf{V}\models u≤ v\vee s{r}^{-1}t\>\iff\>$ | $\displaystyle\mathsf{V}\models{\rm e}≤ v{u}^{-1}\vee s{r}^{-1}t{u}^{-1}$ | |
| --- | --- | --- | --- |
Let us call an ${\mathcal{L}}_{\ell}$ -inequation $\varepsilon$ basic if it is of the form $s≤ t_{1}\vee·s\vee t_{n}$ for an ${\mathcal{L}}_{m}$ -term $s$ and ${\mathcal{L}}_{g}$ -terms $t_{1},...,t_{n}$ . For any basic ${\mathcal{L}}_{\ell}$ -inequation $\varepsilon$ , the ${\mathcal{L}}_{s}$ -inequation $\varepsilon^{\star}$ is defined recursively as follows:
1. If $\varepsilon=(s≤ t_{1}\vee·s\vee t_{n})$ is an ${\mathcal{L}}_{s}$ -inequation, let $\varepsilon^{\star}:=\varepsilon$ ; otherwise, let $i∈\{1,...,n\}$ be minimal such that $t_{i}=u{x}^{-1}v$ for some ${\mathcal{L}}_{m}$ -term $u$ , and let
$$
\varepsilon^{\star}:=(xys\leq xyt_{1}\vee\cdots\vee xyt_{i-1}\vee xyuxs\vee v%
\vee xyt_{i+1}\vee\cdots\vee xyt_{n})^{\star}.
$$
An induction on the number of occurrences in $\varepsilon$ of the inverse operation symbol establishes:
**Proposition 3.7**
*Let $\mathsf{V}$ be any variety of $\ell$ -groups that has the product-splitting property. Then $\mathsf{V}\models\varepsilon\iff\mathsf{V}\models\varepsilon^{\star}$ for any basic ${\mathcal{L}}_{\ell}$ -inequation $\varepsilon$ .*
**Theorem B**
*There are continuum-many equational theories of classes of idempotent semifields.*
**Proof 3.8**
*By Theorem 3.3, there are continuum-many varieties defined relative to the variety of representable $\ell$ -groups by ${\mathcal{L}}_{g}$ -equations, and, by Lemma 3.4, all these varieties have the product-splitting property. For each such variety $\mathsf{V}$ , let $\mathrm{\Sigma}_{\mathsf{V}}$ be the equational theory of the class of the idempotent semiring reducts of the members of $\mathsf{V}$ . Consider now any two distinct varieties $\mathsf{V}_{1}$ and $\mathsf{V}_{2}$ defined relative to the variety of representable $\ell$ -groups by sets of ${\mathcal{L}}_{g}$ -equations. Without loss of generality, there exists a basic ${\mathcal{L}}_{\ell}$ -inequation $\varepsilon$ such that $\mathsf{V}_{1}\models\varepsilon$ and $\mathsf{V}_{2}\not\models\varepsilon$ . But then also $\mathsf{V}_{1}\models\varepsilon^{\star}$ and $\mathsf{V}_{2}\not\models\varepsilon^{\star}$ , by Proposition 3.7, and since $\varepsilon^{\star}$ is an ${\mathcal{L}}_{s}$ -equation, $\mathrm{\Sigma}_{\mathsf{V}_{1}}≠\mathrm{\Sigma}_{\mathsf{V}_{2}}$ . Hence there are continuum-many equational theories of classes of idempotent semifields.*
4 The complexity of the equational theory of the class of idempotent semifields
The equational theory of the variety of Abelian $\ell$ -groups is co-NP-complete [18, Theorem 1], but it follows from the fact that linear programming is solvable in polynomial time that checking if a basic ${\mathcal{L}}_{\ell}$ -inequation is satisfied by all Abelian $\ell$ -groups belongs to P. On the other hand, not only the equational theory of the variety $\mathsf{LG}$ of $\ell$ -groups [5, Theorem 8.3], but also, as we show here, checking if a basic ${\mathcal{L}}_{\ell}$ -inequation is satisfied by $\mathsf{LG}$ , are co-NP-complete, and hence the same is true for the equational theory of the class of idempotent semifields (Theorem C). Indeed, we establish this result by giving a polynomial reduction of the problem of checking the satisfaction of certain ${\mathcal{L}}_{\ell}$ -inequations by $\mathsf{LG}$ that is known to be co-NP-hard, to the problem of checking the satisfaction of certain simple ${\mathcal{L}}_{s}$ -inequations by the class of idempotent semifields.
Let us say that a variety $\mathsf{V}$ of $\ell$ -groups has the meet-splitting property if for any ${\mathcal{L}}_{\ell}$ -terms $s,t,u$ and variable $y$ that does not occur in $s,t,u$ ,
$$
\mathsf{V}\models{\rm e}\leq u\vee(s\wedge t)\iff\mathsf{V}\models{\rm e}\leq u%
\vee sy\vee t{y}^{-1}.
$$
**Lemma 4.1**
*The variety of $\ell$ -groups has the product-splitting and meet-splitting properties.*
**Proof 4.2**
*The fact that $\mathsf{LG}$ has the product-splitting property is established in [3, Lemma 4.1]. To establish the left-to-right direction of the meet-splitting property for $\mathsf{LG}$ , it suffices to show that $\mathsf{LG}\models u\vee(s\wedge t)≤ u\vee sy\vee t{y}^{-1}$ ; just note that for any $\mathbf{L}∈\mathsf{LG}$ and $a,b,c,d∈ L$ , since ${\rm e}≤ d\vee{d}^{-1}$ ,
$$
a\vee(b\wedge c)\leq a\vee(b\wedge c)(d\vee{d}^{-1})=a\vee(b\wedge c)d\vee(b%
\wedge c){d}^{-1}\leq a\vee bd\vee c{d}^{-1}.
$$
For the converse, let $s,t,u$ be any ${\mathcal{L}}_{\ell}$ -terms, let $y$ be a variable that does not occur in $s,t,u$ , and suppose that $\mathsf{LG}\not\models{\rm e}≤ u\vee(s\wedge t)$ . Using lattice-distributivity, we may assume that $\mathsf{LG}\not\models{\rm e}≤ u\vee s$ , the case where $\mathsf{LG}\not\models{\rm e}≤ u\vee t$ being very similar. An ${\mathcal{L}}_{\ell}$ -equation is satisfied by $\mathsf{LG}$ if and only if it is satisfied by the $\ell$ -group $\mathbf{Aut}(\langle{{\mathbb{R}},≤}\rangle)$ consisting of the group of order-preserving bijections of the totally ordered set $\langle{{\mathbb{R}},≤}\rangle$ equipped with the pointwise lattice-order [9, Corollary to Lemma 3]. Hence $\mathbf{Aut}(\langle{{\mathbb{R}},≤}\rangle)\not\models{\rm e}≤ u\vee s$ , and there exists a homomorphism $\varphi\colon\mathbf{Tm}({\mathcal{L}}_{\ell})→\mathbf{Aut}(\langle{{\mathbb%
{R}},≤}\rangle)$ and $q∈{\mathbb{R}}$ such that $(q)\varphi_{u}<q$ and $(q)\varphi_{s}<q$ , where we assume that order-preserving bijections act on $\langle{{\mathbb{R}},≤}\rangle$ from the right, and write $\varphi_{v}$ for $\varphi(v)$ . We obtain a homomorphism $\hat{\varphi}\colon\mathbf{Tm}({\mathcal{L}}_{\ell})→\mathbf{Aut}(\langle{{%
\mathbb{R}},≤}\rangle)$ by defining $\hat{\varphi}_{x}:=\varphi_{x}$ for every variable $x≠y$ and defining $\hat{\varphi}_{y}$ such that $(q)\varphi_{s}\hat{\varphi}_{y}=(q)\varphi_{s}<q$ and $(q)\varphi_{t}<(q)\hat{\varphi}_{y}$ . Note that such a definition of $\hat{\varphi}_{y}$ is possible because $(q)\hat{\varphi}_{y}$ can be chosen to be arbitrarily large and any partial order-preserving injective map on $\langle{{\mathbb{R}},≤}\rangle$ extends linearly to a member of $\mathbf{Aut}(\langle{{\mathbb{R}},≤}\rangle)$ . It follows, since $y$ does not occur in $s,t,u$ , that $(q)\hat{\varphi}_{u}=(q)\varphi_{u}<q$ , $(q)\hat{\varphi}_{s}\hat{\varphi}_{y}=(q)\varphi_{s}\hat{\varphi}_{y}=(q)%
\varphi_{s}<q$ , and $(q)\hat{\varphi}_{t}=(q)\varphi_{t}<(q)\hat{\varphi}_{y}$ . Hence $\mathsf{LG}\not\models{\rm e}≤ u\vee sy\vee t{y}^{-1}$ .*
**Remark 4.3**
*No non-trivial proper subvariety of $\mathsf{LG}$ has the meet-splitting property. It follows easily from [14, Example 13] that the ${\mathcal{L}}_{\ell}$ -equation ${\rm e}≤(x\vee{\rm e})^{2}{z}^{-1}\vee{(x\vee{\rm e})}^{-1}z\vee{(x\vee{\rm
e%
})}^{-1}$ is satisfied by every proper subvariety of $\mathsf{LG}$ ; indeed, it axiomatizes the variety of normal-valued $\ell$ -groups, the unique co-atom in the subvariety lattice of $\ell$ -groups, relative to $\mathsf{LG}$ . Hence, if a proper subvariety of $\mathsf{LG}$ has the meet-splitting property, it satisfies also ${\rm e}≤{(x\vee{\rm e})}^{-1}$ and is trivial.*
**Proposition 4.4**
*The problem of checking if a basic ${\mathcal{L}}_{\ell}$ -inequation is satisfied by the variety of $\ell$ -groups is co-NP-complete.*
**Proof 4.5**
*Since checking if an ${\mathcal{L}}_{\ell}$ -inequation is satisfied by $\mathsf{LG}$ belongs to co-NP [5, Theorem 8.3], it suffices to present a polynomial time algorithm that given input with an ${\mathcal{L}}_{\ell}$ -equation $\varepsilon$ of the form
$$
\bigwedge_{i\in I}\bigvee_{j\in J_{i}}s_{ij}\leq\bigvee_{k\in K}\bigwedge_{l%
\in L_{k}}t_{kl}\vee u_{kl},
$$
where each $s_{ij}$ , $t_{kl}$ , and $u_{kl}$ is an ${\mathcal{L}}_{g}$ -term, outputs a basic ${\mathcal{L}}_{\ell}$ -inequation $\delta$ that has size polynomial in the size of $\varepsilon$ such that $\mathsf{LG}\models\varepsilon\iff\mathsf{LG}\models\delta$ . This is a consequence of the fact that the problem of checking validity in $\mathsf{LG}$ of such ${\mathcal{L}}_{\ell}$ -equations is co-NP-hard, since this is the case even for distributive lattice equations of this form, where each $s_{ij}$ , $t_{kl}$ , and $u_{kl}$ is a variable [10, Corollary 2.7]. First, we do a little preprocessing, using the fact that $\mathsf{LG}$ has the product-splitting property for the third equivalence:
| | $\displaystyle\mathsf{LG}\models\varepsilon$ | $\displaystyle\iff\mathsf{LG}\models{\rm e}≤(\bigvee_{k∈ K}\bigwedge_{l∈
L%
_{k}}t_{kl}\vee u_{kl}){(\bigwedge_{i∈ I}\bigvee_{j∈ J_{i}}s_{ij})}^{-1}$ | |
| --- | --- | --- | --- |
Hence we may assume without loss of generality that $\varepsilon$ is an ${\mathcal{L}}_{\ell}$ -equation of the form ${\rm e}≤ u_{1}\vee·s\vee u_{n}$ , where each $u_{i}$ is a meet of binary joins of ${\mathcal{L}}_{g}$ -terms, and let $S$ be the size of $\varepsilon$ , so that at most $S$ ${\mathcal{L}}_{g}$ -terms and at most $S$ meets occur in $\varepsilon$ . If $\varepsilon$ contains no meets, it is a basic ${\mathcal{L}}_{\ell}$ -inequation of the required form. Otherwise, suppose that $u_{1}=s_{1}\wedge·s\wedge s_{m}$ such that $s_{i}=s_{i1}\vee s_{i2}$ and let $u:=u_{2}\vee·s\vee u_{n}$ . By choosing distinct variables $y_{1},...,y_{m}$ that do not occur in $\varepsilon$ and using the meet-splitting property repeatedly,
| | $\displaystyle\mathsf{LG}\models\varepsilon$ | $\displaystyle\iff\mathsf{LG}\models{\rm e}≤ u\vee(s_{1}\wedge·s\wedge s%
_{m})$ | |
| --- | --- | --- | --- |
Let $\varepsilon^{\prime}$ be the ${\mathcal{L}}_{\ell}$ -equation ${\rm e}≤ u\vee s_{11}y_{1}\vee s_{12}y_{1}\vee(\bigvee_{i=2}^{m}s_{i1}{y}^{%
-1}_{1}·s{y}^{-1}_{i-1}y_{i}\vee s_{i2}{y}^{-1}_{1}·s{y}^{-1}_{i-1}y_{%
i})$ , observing that $\varepsilon^{\prime}$ has the same number of ${\mathcal{L}}_{g}$ -terms as $\varepsilon$ , and that each of these ${\mathcal{L}}_{g}$ -terms has size at most $2S$ . Hence, repeating this procedure for $u_{2},...,u_{n}$ , we obtain in polynomial time a basic ${\mathcal{L}}_{\ell}$ -inequation $\delta$ of size at most $2S^{2}$ such that $\mathsf{LG}\models\varepsilon\iff\mathsf{LG}\models\delta$ .*
**Proposition 4.6**
*The problem of checking if a simple ${\mathcal{L}}_{s}$ -inequation is satisfied by the class of semifields (or, equivalently, by the variety of $\ell$ -groups) is co-NP-complete.*
**Proof 4.7**
*Recall that $\mathsf{LG}$ has the product-splitting property, by Lemma 4.1. Hence, it suffices, by Propositions 4.4 and 3.7, to present a polynomial time algorithm that given a basic ${\mathcal{L}}_{\ell}$ -inequation $\varepsilon$ of the form $s≤ t_{1}\vee·s\vee t_{k}$ as input produces the simple ${\mathcal{L}}_{s}$ -inequation $\varepsilon^{\star}$ as output, where the size of $\varepsilon^{\star}$ is polynomial in the size of $\varepsilon$ . Let $S$ be the size of $\varepsilon$ and recall the recursive definition of $\varepsilon^{\star}$ from Section 3. The number of inverses in $\varepsilon$ is bounded by $S$ and decreases in every step of the recursive definition, so the algorithm stops after at most $S$ steps and yields the ${\mathcal{L}}_{s}$ -inequation $\varepsilon^{\star}$ . Moreover, each step increases the length of the ${\mathcal{L}}_{m}$ -term on the left of the basic ${\mathcal{L}}_{\ell}$ -inequation by $2$ and increases by $1$ the number of ${\mathcal{L}}_{g}$ -terms on the right. The length of the ${\mathcal{L}}_{m}$ -term on the left is therefore at most $3S$ and the number of ${\mathcal{L}}_{g}$ -terms is at most $2S$ in every step. It follows that in every step the size of the ${\mathcal{L}}_{s}$ -inequation is increased by at most $3S+2· 2S=7S$ and hence $\varepsilon^{\star}$ is of size at most $7S^{2}+S$ . It is also clear that $\varepsilon^{\star}$ can be computed from $\varepsilon$ in polynomial time.*
As an immediate consequence of Proposition 4.6 we obtain the main result of this section.
**Theorem C**
*The equational theory of the class of idempotent semifields is co-NP-complete.*
We conclude this section by using Theorem C and a correspondence established in [4] to prove complexity results also for the existence of right orders on free groups and monoids satisfying finitely many constraints. Recall first that a right order on a monoid $\mathbf{M}$ is a total order $≤$ on $M$ such that $a≤ b\>\Longrightarrow\>ac≤ bc$ for any $a,b,c∈{M}$ . For a set $X$ we let $\mathbf{F}_{g}(X)$ and $\mathbf{F}_{m}(X)$ denote the free group and free monoid with generators in $X$ , respectively, assuming for convenience that ${\rm F}_{g}(X)⊂eq{\rm Tm}({\mathcal{L}}_{g})$ and ${\rm F}_{m}(X)⊂eq{\rm Tm}({\mathcal{L}}_{m})$ .
**Theorem 4.8 ([4, Theorem 2])**
*For any set $X$ and $s_{1},...,s_{n}∈{\rm F}_{g}(X)$ , there exists a right order $≤$ on $\mathbf{F}_{g}(X)$ satisfying ${\rm e}<s_{1},...,{\rm e}<s_{n}$ if and only if $\mathsf{LG}\not\models{\rm e}≤ s_{1}\vee·s\vee s_{n}$ .*
**Corollary 4.9**
*The problem of checking for a set $X$ with $\lvert X\rvert≥ 2$ and $s_{1},...,s_{n}∈{\rm F}_{g}(X)$ if there exists a right order $≤$ on $\mathbf{F}_{g}(X)$ satisfying ${\rm e}<s_{1},...,{\rm e}<s_{n}$ is NP-complete.*
**Proof 4.10**
*Observe first that when $X$ is an infinite set, the claim follows directly from Proposition 4.4 and Theorem 4.8. To establish the claim in full generality, it suffices, by Theorem 4.8, to consider the case where $\lvert X\rvert=2$ . Let $X:=\{x_{1},x_{2}\}$ and $Y:=\{y_{n}\mid n∈{\mathbb{N}}\}$ . It is well-known that $\mathbf{F}_{g}(Y)$ is isomorphic to the commutator subgroup $\mathbf{G}$ of $\mathbf{F}_{g}(X)$ generated by elements of the form $[x_{1}^{k},x_{2}^{l}]$ with $k,l∈{\mathbb{Z}}{\setminus}\{0\}$ (see, e.g., [17, Theorem 11.48]). In particular, for any bijection $\pi\colon({\mathbb{Z}}{\setminus}\{0\})^{2}→{\mathbb{N}}$ , we can define an isomorphism $\varphi\colon\mathbf{F}_{g}(Y)→\mathbf{G}$ such that $\varphi(y_{\pi(\langle{k,l}\rangle)}):=[x_{1}^{k},x_{2}^{l}]$ for $k,l∈{\mathbb{Z}}{\setminus}\{0\}$ . Moreover, since $\mathbf{F}_{g}(X)/\mathbf{G}\cong\langle{{\mathbb{Z}}^{2},+,-,\langle{0,0}%
\rangle}\rangle$ , there exists a right order $\preceq$ on the group $\mathbf{F}_{g}(X)/\mathbf{G}$ . Hence, for any right order $≤$ on $\mathbf{G}$ , we obtain a right order $≤^{\ast}$ on $\mathbf{F}_{g}(X)$ that extends $≤$ by setting
$$
s\leq^{\ast}t\>:\Longleftrightarrow\>Gs\prec Gt\,\text{ or }\,(Gs=Gt\,\text{ %
and }\,s\leq t).
$$
It follows that there exists a right order on $\mathbf{F}_{g}(Y)$ satisfying ${\rm e}<s_{1},...,{\rm e}<s_{n}$ , for some given $s_{1},...,s_{n}∈{\rm F}_{g}(Y)$ , if and only if there exists a right order on $\mathbf{F}_{g}(X)$ satisfying ${\rm e}<\varphi(s_{1}),...,{\rm e}<\varphi(s_{n})$ . Finally, note that $\pi$ can be chosen such that $\varphi(s_{i})$ is computable in polynomial time from $s_{i}$ for each $i∈\{1,...,n\}$ , and its size is polynomial in the sum of the sizes of $s_{1},...,s_{n}$ .*
By [3, Corollary 3.4], every right order on $\mathbf{F}_{m}(X)$ extends to a right order on $\mathbf{F}_{g}(X)$ . Hence it follows from Theorem 4.8 that $\mathsf{LG}\not\models s≤ t_{1}\vee·s\vee t_{n}$ , where $s,t_{1},...,t_{n}∈{\rm F}_{m}(X)$ , if and only if there exists a right order $≤$ on $\mathbf{F}_{m}(X)$ with $s<t_{1},...,s<t_{n}$ . Proposition 4.6 therefore implies that the problem of checking for any $s,t_{1},...,t_{n}∈{\rm F}_{m}(\omega)$ if there exists a right order $≤$ on $\mathbf{F}_{m}(\omega)$ satisfying $s<t_{1},...,s<t_{n}$ is NP-complete. Therefore:
**Corollary 4.11**
*The problem of checking for any $s_{1},...,s_{n},t_{1},...,t_{n}∈{\rm F}_{m}(\omega)$ if there exists a right order $≤$ on $\mathbf{F}_{m}(\omega)$ satisfying $s_{1}<t_{1},...,s_{n}<t_{n}$ is NP-complete.*
Note that it does not follow directly from the previous results that Corollary 4.11 is true when $\mathbf{F}_{m}(\omega)$ is replaced by $\mathbf{F}_{m}(X)$ for a set $X$ with $2≤\lvert X\rvert<\omega$ , the main obstacle being that the translation $\varphi$ in the proof of Corollary 4.9 introduces new inverses, while the elimination of inverses in the proof of Proposition 4.6 introduces new variables.
5 Related structures
In this final section, we show that the results of the previous sections extend in many cases to other classes of algebraic structures that are closely related to idempotent semifields and $\ell$ -groups.
Let us remark first that expanding any idempotent semifield with the lattice meet operation produces a distributive $\ell$ -monoid: an algebraic structure $\langle{L,\wedge,\vee,·,{\rm e}}\rangle$ such that $\langle{L,·,{\rm e}}\rangle$ is a monoid; $\langle{L,\wedge,\vee}\rangle$ is a distributive lattice; and multiplication distributes over binary meets and joins. It follows directly from Theorem B that there are continuum-many equational theories of classes of distributive $\ell$ -monoids with an idempotent semifield reduct, and from Theorem C, that the equational theory of the class of distributive $\ell$ -monoids with an idempotent semifield reduct is co-NP-complete. Although not all distributive $\ell$ -monoids are meet-expansions of idempotent semifields (equivalently, inverse-free reducts of $\ell$ -groups), the equational theories of the classes of distributive $\ell$ -monoids and meet-expansions of idempotent semifields (equivalently, inverse-free reducts of $\ell$ -groups) coincide [3, Theorem 2.9]. Hence the equational theory of the class of meet-expansions of idempotent semifields is finitely based and an analogue of Theorem A does not hold in this setting. Note, however, that even though the equational theory of the variety of Abelian $\ell$ -groups is finitely based, this is not the case for the class of their inverse-free reducts [16, Theorem 2].
Recall next that idempotent semifields are sometimes formulated in the literature without the neutral element ${\rm e}$ in the signature, that is, as ${\rm e}$ -free reducts of idempotent semifields as defined in this paper. Observe, however, that a simple ${\mathcal{L}}_{s}$ -inequation $s≤ t_{1}\vee·s\vee s_{n}$ is satisfied by an idempotent semifield $\mathbf{S}$ if and only if its ${\rm e}$ -free reduct satisfies $(xs)^{\circ}≤(xt_{1})^{\circ}\vee·s\vee(xs_{n})^{\circ}$ , where $x$ is any variable not occurring in $s,t_{1},...,t_{n}$ , and $v^{\circ}$ is obtained by removing all occurrences of ${\rm e}$ from an ${\mathcal{L}}_{s}$ -term $v$ . Hence, we obtain easily the following analogues of Theorems A, B, and C: there is no non-trivial class of ${\rm e}$ -free reducts of idempotent semifields that is finitely based, there are continuum-many equational theories of classes of ${\rm e}$ -free reducts of idempotent semifields, and the equational theory of the class of ${\rm e}$ -free reducts of idempotent semifields is co-NP-complete.
Finally, recall that idempotent semirings are also sometimes formulated in the literature with both ${\rm e}$ and a constant symbol $0 0$ interpreted as the neutral element of $\vee$ . We show here that analogues of our Theorems A, B, and C also hold in this setting, using similar methods to [1, Section 4]. Let us call an algebraic structure $\langle{F,\vee,·,{\rm e},0}\rangle$ an idempotent $0 0$ -semiring if $\langle{F,\vee,·,{\rm e}}\rangle$ is an idempotent semiring with least element $0 0$ , and an idempotent 0-semifield if, additionally, $\langle{F{\setminus}\{0\},·,{\rm e}}\rangle$ is the monoid reduct of a group. Clearly, if $\mathbf{F}=\langle{F,\vee,·,{\rm e},0}\rangle$ is an idempotent $0 0$ -semifield, then $\mathbf{F}^{*}:=\langle{F{\setminus}\{0\},\vee,·,{\rm e}}\rangle$ is an idempotent semifield. Conversely, given any idempotent semiring $\mathbf{S}=\langle{S,\vee,·,{\rm e}}\rangle$ and element $0\not∈ S$ , the algebraic structure $\mathbf{S}_{0}:=\langle{S\cup\{0\},\vee,·,{\rm e},0}\rangle$ satisfying $0≤ a$ and $0· a=a· 0=0$ for all $a∈ S\cup\{0\}$ , is an idempotent $0 0$ -semiring. In particular, if $\mathbf{S}$ is an idempotent semifield, then $\mathbf{S}_{0}$ is an idempotent $0 0$ -semifield. Moreover, $(\mathbf{F}^{*})_{0}=\mathbf{F}$ for each idempotent $0 0$ -semifield $\mathbf{F}$ , and $(\mathbf{S}_{0})^{*}=\mathbf{S}$ for each idempotent semifield $\mathbf{S}$ . Hence there is a one-to-one correspondence between classes of idempotent $0 0$ -semifields and classes of idempotent semifields implemented by the maps $\mathsf{K}\mapsto\{\mathbf{F}^{*}\mid\mathbf{F}∈\mathsf{K}\}$ and $\mathsf{K}\mapsto\{\mathbf{S}_{0}\mid\mathbf{S}∈\mathsf{K}\}$ .
**Remark 5.1**
*There is a linear time algorithm that given any term $t$ in the signature of idempotent $0 0$ -semirings, produces a smaller term $t^{\prime}$ that is either $0 0$ or an ${\mathcal{L}}_{s}$ -term such that $t≈ t^{\prime}$ is satisfied by all idempotent $0 0$ -semirings that satisfy the absorption laws $x· 0≈ 0$ and $0· x≈ 0$ (in particular, all idempotent $0 0$ -semifields).*
Let us call a simple ${\mathcal{L}}_{s}$ -inequation $s≤ t_{1}\vee·s\vee t_{n}$ right-regular if every variable occurring in $t_{1},...,t_{n}$ occurs in $s$ .
**Lemma 5.2**
*Let $\mathbf{S}$ be any idempotent semiring. Then a right-regular simple ${\mathcal{L}}_{s}$ -inequation is satisfied by $\mathbf{S}$ if and only if it is satisfied by $\mathbf{S}_{0}$ .*
**Proof 5.3**
*The right-to-left direction follows from the fact that $\mathbf{S}$ is a subreduct of $\mathbf{S}_{0}$ . For the left-to-right direction, it suffices to observe that a right-regular simple ${\mathcal{L}}_{s}$ -inequation is satisfied by $\mathbf{S}_{0}$ under any assignment that maps one of the variables occurring in it to $0 0$ .*
**Lemma 5.4**
*Let $\mathbf{S}$ be any non-trivial idempotent semifield. Then for any simple ${\mathcal{L}}_{s}$ -inequation $\varepsilon=(s≤ t_{1}\vee·s\vee t_{n})$ satisfied by $\mathbf{S}$ , there exists a subset $\{t_{i_{1}},...,t_{i_{k}}\}⊂eq\{t_{1},...,t_{n}\}$ such that the simple ${\mathcal{L}}_{s}$ -inequation $s≤ t_{i_{1}}\vee·s\vee t_{i_{k}}$ is right-regular and $\mathbf{S}\models s≤ t_{i_{1}}\vee·s\vee t_{i_{k}}$ .*
**Proof 5.5**
*Note first that, since $\mathbf{S}$ is non-trivial, there cannot be a variable $x$ that occurs in each of $t_{1},...,t_{n}$ but not in $s$ ; otherwise, we could assign $x$ to some element $a<{\rm e}$ in $S$ and all other variables to ${\rm e}$ to arrive at a contradiction. Hence it suffices to prove that (assuming a suitable permutation of $t_{1},...,t_{n}$ ) if a variable $x$ occurs in each of $t_{k+1},...,t_{n}$ , but not in $s,t_{1},...,t_{k}$ for some $k<n$ , then $\mathbf{S}$ satisfies $s≤ t_{1}\vee·s\vee t_{k}$ . In this way we can inductively eliminate all the terms on the right that contain a variable that does not occur in $s$ and obtain a right-regular simple ${\mathcal{L}}_{s}$ -inequation. Suppose contrapositively that $\mathbf{S}\not\models s≤ t_{1}\vee·s\vee t_{k}$ , that is, there exists a homomorphism $\varphi\colon\mathbf{Tm}({\mathcal{L}}_{s})→\mathbf{S}$ such that $\varphi(s)\not≤\varphi(t_{1})\vee·s\vee\varphi(t_{k})$ . We may assume that for $i∈\{k+1,...,n\}$ , each $t_{i}$ is of the form $u_{1}xu_{2}x·s u_{l-1}xu_{l}$ , where $u_{1},...,u_{l}$ do not contain $x$ , by inserting ${\rm e}$ where needed. Let $\hat{\varphi}\colon\mathbf{Tm}({\mathcal{L}}_{s})→\mathbf{S}$ be the homomorphism defined by setting $\hat{\varphi}(y):=\varphi(y)$ for every variable $y≠x$ and $\hat{\varphi}(x)$ to be the meet of all the elements of the form ${\varphi(u)}^{-1}\wedge{\varphi(u)}^{-1}\varphi(t_{1}){\varphi(v)}^{-1}$ such that $u,v$ are subterms of $t_{k+1},...,t_{n}$ not containing $x$ . Then for each $i∈\{k+1,...,n\}$ and $t_{i}=u_{1}xu_{2}x·s u_{l-1}xu_{l}$ , where $u_{1},...,u_{l}$ do not contain $x$ ,
| | $\displaystyle\hat{\varphi}(t_{i})$ | $\displaystyle=\varphi(u_{1})\hat{\varphi}(x)\varphi(u_{2})\hat{\varphi}(x)%
·s\varphi(u_{l-1})\hat{\varphi}(x)\varphi(u_{l})$ | |
| --- | --- | --- | --- |
Hence $\hat{\varphi}(t_{k+1})\vee·s\vee\hat{\varphi}(t_{n})≤\varphi(t_{1})$ , so $\hat{\varphi}(s)=\varphi(s)\not≤\varphi(t_{1})\vee·s\vee\varphi(t_{k})=%
\hat{\varphi}(t_{1})\vee·s\vee\hat{\varphi}(t_{n})$ , and $\mathbf{S}\not\models s≤ t_{1}\vee·s\vee t_{n}$ .*
The equational theory of any non-empty class of idempotent $0 0$ -semifields containing exactly two elements has a finite basis consisting of the defining equations for bounded distributive lattices with meet operation $·$ , greatest element ${\rm e}$ , and least element $0 0$ . For convenience, let us call a class of idempotent $0 0$ -semifields non-Boolean if at least one of its members has more than two elements. Clearly, a class $\mathsf{K}$ of idempotent $0 0$ -semifields is non-Boolean if and only if $\mathsf{K}^{*}$ is non-trivial.
**Corollary 5.6**
*Every non-Boolean class of idempotent $0 0$ -semifields is not finitely based.*
**Proof 5.7**
*Suppose towards a contradiction that $\mathsf{K}$ is a finitely based non-Boolean class of idempotent $0 0$ -semifields. Using Remark 5.1 and Lemma 5.4, we may assume that $\mathrm{\Sigma}\cup\{0≤ x,x· 0≈ 0,0· x≈ 0\}$ is a basis for ${\rm Eq}(\mathsf{K})$ for some finite set of right-regular simple ${\mathcal{L}}_{s}$ -inequations $\mathrm{\Sigma}$ . That is, $\mathrm{\Sigma}\cup\{0≤ x,x· 0≈ 0,0· x≈ 0\}⊂eq{%
\rm Eq}(\mathsf{K})$ and ${\rm Eq}(\mathsf{K})$ is a logical consequence of $\mathrm{\Sigma}\cup\{0≤ x,x· 0≈ 0,0· x≈ 0\}$ . We claim that $\mathrm{\Sigma}$ is a basis for ${\rm Eq}(\mathsf{K}^{*})$ , contradicting Theorem A. Observe first that $\mathsf{K}\models\varepsilon$ if and only if $\mathsf{K}^{*}\models\varepsilon$ , for any right-regular simple ${\mathcal{L}}_{s}$ -inequation $\varepsilon$ , by Lemma 5.2, so $\mathrm{\Sigma}⊂eq{\rm Eq}(\mathsf{K}^{*})⊂eq{\rm Eq}(\mathsf{K})$ . Now suppose that some idempotent semiring $\mathbf{A}$ satisfies $\mathrm{\Sigma}$ . Then $\mathbf{A}_{0}$ satisfies $\mathrm{\Sigma}$ , by Lemma 5.2, and $\mathbf{A}_{0}$ therefore satisfies ${\rm Eq}(\mathsf{K})$ . So $\mathbf{A}$ satisfies ${\rm Eq}(\mathsf{K}^{*})$ . Hence ${\rm Eq}(\mathsf{K}^{*})$ is a logical consequence of $\mathrm{\Sigma}$ .*
If $\mathsf{K}$ and $\mathsf{K^{\prime}}$ are classes of idempotent semifields with distinct equational theories, then there is a simple ${\mathcal{L}}_{s}$ -inequation that is satisfied by one and not the other, so $\mathsf{K}_{0}$ and $\mathsf{K}^{\prime}_{0}$ also have distinct equational theories, by Lemma 5.2 and Lemma 5.4. Hence, by Theorem B:
**Corollary 5.8**
*There are continuum-many equational theories of classes of idempotent $0 0$ -semifields.*
Finally, combining Remark 5.1, Lemma 5.2, Lemma 5.4, and Theorem C, we obtain:
**Corollary 5.9**
*The equational theory of the class of idempotent $0 0$ -semifields is co-NP-complete.*
References
- [1] L. Aceto, Z. Ésik, and A. Ingólfsdóttir. Equational theories of tropical semirings. Theor. Comput. Sci., 298(3):417–469, 2003.
- [2] M.E. Anderson and T.H. Feil. Lattice-Ordered Groups: An Introduction. Springer, 1988.
- [3] A. Colacito, N. Galatos, G. Metcalfe, and S. Santschi. From distributive $\ell$ -monoids to $\ell$ -groups, and back again. J. Algebra, 601:129–148, 2022.
- [4] A. Colacito and G. Metcalfe. Ordering groups and validity in lattice-ordered groups. J. Pure Appl. Algebra, 223(12):5163–5175, 2019.
- [5] N. Galatos and G. Metcalfe. Proof theory for lattice-ordered groups. Ann. Pure Appl. Logic, 8(167):707–724, 2016.
- [6] J.S. Golan. Semirings and their Applications. Kluwer, 1999.
- [7] J. Gunawardena, editor. Idempotency, volume 11 of Publications of the Newton Institute. Cambridge University Press, 1998.
- [8] U. Hebisch and H.J. Weinert. Semi-rings and semi-fields. In M. Hazewinkel, editor, Handbook of Algebra, volume 1, pages 427–462. North Holland, 1995.
- [9] W.C. Holland. The largest proper variety of lattice-ordered groups. Proc. Amer. Math. Soc., 57:25–28, 1976.
- [10] H.B. Hunt III, D.J. Rosenkrantz, and P.A. Bloniarz. On the computational complexity of algebra on lattices. SIAM J. Comput., 16(1):129–148, 1987.
- [11] M. Jackson. Flat algebras and the translation of universal horn logic to equational logic. J. Symbolic Logic, 73(1):90–128, 2008.
- [12] M. Jackson, M. Ren, and X. Zhao. Nonfinitely based ai-semirings with finitely based semigroup reducts. J. Algebra, 611:211–245, 2022.
- [13] V.M. Kopytov and N.Y. Medvedev. Varieties of lattice-ordered groups. Algebra and Logic, 16:281–285, 1977.
- [14] S. McCleary. The word problem in free normal valued lattice-ordered groups: a solution and practical shortcuts. Algebra Universalis, 14:317–348, 1982.
- [15] G. Metcalfe, F. Paoli, and C. Tsinakis. Residuated Structures in Algebra and Logic, volume 277 of Mathematical Surveys and Monographs. American Mathematical Society, 2023.
- [16] V. B. RepnitskiÄ. Bases of identities of varieties of lattice-ordered semigroups. Algebra i Logika, 22(6):649–665, 720, 1983.
- [17] J.J. Rotman. An Introduction to the Theory of Groups, volume 148 of Graduate Texts in Mathematics. Springer, 4 edition, 1994.
- [18] V. Weispfenning. The complexity of the word problem for Abelian $l$ -groups. Theor. Comput. Sci., 48:127–132, 1986.
G. Metcalfe and S. Santschi Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern Switzerland