Ideals and their complements in commutative semirings

We study conditions under which the lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{\mathbf {Id}}}}\mathbf R$$\end{document}IdR of ideals of a given a commutative semiring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}$$\end{document}R is complemented. At first we check when the annihilator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^*$$\end{document}I∗ of a given ideal I of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {R}}$$\end{document}R is a complement of I. Further, we study complements of annihilator ideals. Next we investigate so-called Łukasiewicz semirings. These form a counterpart to MV-algebras which are used in quantum structures as they form an algebraic semantic of many-valued logics as well as of the logic of quantum mechanics. We describe ideals and congruence kernels of these semirings with involution. Finally, using finite unitary Boolean rings, a construction of commutative semirings with complemented lattice of ideals is presented.


Introduction
Semirings play an important role in both algebra and applications. They share several important properties of rings, but, on the other hand, every distributive lattice with the least element can be recognized as an idempotent semiring. Hence, if the addition operation of a semiring is idempotent, then the semiring often shares some properties with semilattices or lattices. Contrary to rings, ideals of semirings need not be zero-classes of congruences, so-called congruence kernels. However, the set of all ideals forms a complete lattice similarly as for rings. We study under what conditions this lattice is complemented. It turns out that for commutative semirings having no 2-nilpotent element, one complement of a given ideal is given by its annihilator. Analogous results are obtained for the lattice of annihilator ideals though this lattice need not be a sublattice of the lattice of ideals.
In the second part we study certain semirings with involution, so-called Łukasiewicz semirings and their ideals. These semirings originated in the study of quantum structures, see, for example, Bonzio et al. (2016) and Chajda et al. (2018) for the concepts and motivation. The ideals of Łukasiewicz semirings have interesting properties, and the congruence kernels of these semirings can be described easily. Complements within the lattice of ideals can be described by means of complemented elements.
It might be the case that commutative semirings are complemented but their complements do not coincide with the corresponding annihilators. We show that by forming the direct product of such semiring with a finite unitary Boolean ring we obtain a commutative semiring whose lattice of ideals has similar properties as the lattice of ideals of the original semiring. Here we use the fact that in such a case we have no skew ideals in the direct product and hence the lattice of ideals of the new semiring is the direct product of the lattices of ideals of its factors.

Basic concepts
In the literature there exist different definitions of a semiring. We will use the following one which differs from that in Golan (1999) (where it is called a "hemiring").
A commutative semiring is an algebra R = (R, +, ·, 0) of type (2, 2, 0) such that If there exists an element 1 of R satisfying the identity x ·1 ≈ x then R is called unitary, and if R satisfies the identity For a ∈ R we write a A instead of {a}A etc.
An ideal of R is a subset I of R satisfying 0 ∈ I , I + I ⊆ I and I R ⊆ I . Let Id R denote the set of all ideals of R. Since Id R is closed with respect to arbitrary set-theoretical intersections, where s∈S I s denotes the set of all sums of finitely many elements of s∈S I s .
In the following we will write x y instead of x · y and we will denote the set of all nonnegative integers by N 0 .
In order to study the structure of lattices of ideals, we remember several concepts from lattice theory.
Recall that a lattice (L, ∨, ∧) is called modular if it satisfies the identity Fig. 1 The lattice of ideals of R from Example 2.2 It is well known that lattices of ideals of rings are modular. Unfortunately, this need not hold for lattices of ideals of semirings, as the following example shows. Obviously, for an arbitrary subset A of R the set I (A) := {a 1 r 1 + · · · + a k r k + n 1 a k+1 + · · · + n l a k+l | k, l ≥ 0, a 1 , . . . , a k+l ∈ A, r 1 , . . . , r k ∈ R, n 1 , . . . , n l ∈ N 0 } is the ideal generated by A. Here, for arbitrary a ∈ R we have 0a := 0, 1a := a, 2a := a + a, . . . .
Particularly, for a ∈ R the set I (a) := a R + N 0 a (where N 0 a denotes the set {na | n ∈ N 0 }) is the ideal generated by a. Such ideals are called principal.

Annihilators as complements of ideals
For any I ∈ Id R we define I * := {x ∈ R | x I = {0}}, call I * an annihilator ideal of R and * annihilation or annihilation mapping and put Ann R := {I * | I ∈ Id R}. It is easy to see that Ann R ⊆ Id R. Moreover, Ann R := (Ann R, ⊆) is a complete lattice. This follows from (i) of Lemma 3.3.
The following facts are straightforward.
In the following we are going to investigate the lattice operations in Id R and Ann R. Since these lattices are complete, we consider, more generally, infinite joins and meets. Lemma 3.3 Let R = (R, +, ·, 0) be a commutative semiring. In Id R we have On the other hand, if a ∈ s∈S I * s and b ∈ s∈S I s then there exist b 1 , . . . , b n ∈ s∈S I s with b 1 + · · · + b n = b and hence ab = a(b 1 + · · · + b n ) = ab 1 + · · · + ab n = 0 + · · · + 0 = 0, i.e. a ∈ ( s∈S I s ) * showing s∈S I * s ⊆ ( s∈S I s ) * and hence (i).
(ii) We have s∈S I s ⊆ I t for all t ∈ S and hence I * t ⊆ ( s∈S I s ) * for all t ∈ S which shows (ii).
As remarked above, since in Ann R we have Ann R is a complete lattice.
The following lemma shows that in general the supremum within Ann R may differ from that within Id R.
Proof For all t ∈ S we have I t ⊆ s∈S I s and hence Moreover, if I ∈ Ann R and I s ⊆ I for all s ∈ S then s∈S I s ⊆ I and hence I * ⊆ ( s∈S I s ) * whence ( s∈S I s ) * * ⊆ I * * = I . This proves (2). Finally, according to (1), Ann R is closed under arbitrary intersections which yields (3).
The lattice of ideals and the annihilator lattice of R can be written in the form Now, we define the concept which plays a crucial role for complementation in Id R.
In the following we set and and investigate under which conditions these algebras are ortholattices. Obviously, Lemma 3.6 Let R = (R, +, ·, 0) be a commutative semiring. Then for every ideal I of R its pseudocomplement in Id R is just I * if and only if R has no 2-nilpotent element.
Proof First assume I * to be the pseudocomplement of I in Id R for each I ∈ Id R. If a ∈ R and a 2 = 0 then a ∈ I (a)∩I (a) * = {0} and hence a = 0. Conversely, assume R to for all j ∈ J and k ∈ K and hence K ⊆ J * . This shows that J * is the pseudocomplement of J in Id R completing the proof of the lemma.
Theorem 3.7 Let R = (R, +, ·, 0) be a commutative semiring. Then Id * R is an ortholattice if and only if I * * = I for all I ∈ Id R and R has no 2-nilpotent element.
Proof First assume Id * R to be an ortholattice. Then I * * = I for all I ∈ Id R. Moreover, if a ∈ R and a 2 = 0 then a ∈ I (a) ∩ I (a) * = {0} and hence a = 0. Conversely, assume I * * = I for all I ∈ Id R and R to have no 2-nilpotent element. Let J ∈ Id R. According to Lemma 3.6, J ∩ J * = {0}. Now, according to (i) of Lemma 3.3, Finally, the de Morgan laws hold because of Lemma 3.2.
The following two examples show which role the existence of 2-nilpotent elements plays for the fact of annihilators to be complements.
It is easy to check that R is not a ring. R has the fol-  Fig. 3. The element b is 2-nilpotent and hence, according to Theorem 3.7, Id * R is not an ortholattice as can be seen from It is easy to see that {0} * * = {0} if and only if there exists some x ∈ R\{0} with x R = {0}. Hence, if there exists such an element x, then Id * R is not an ortholattice.
If Ann * R is considered instead of Id * R, then the condition for being an ortholattice can be simplified as follows: Theorem 3.10 Let R = (R, +, ·, 0) be a commutative semiring. Then Ann * R is an ortholattice if and only if there exists no a ∈ R\R * with a 2 = 0.
Proof First assume Ann * R to be an ortholattice. If a ∈ R and a 2 = 0 then a ∈ I (a) * ∩ I (a) * * = R * . Conversely, assume that there exists no b ∈ R\R * with b 2 = 0. Let J ∈ Ann R. If c ∈ J ∩ J * then c 2 = 0 whence c ∈ R * which shows J ∩ J * = R * . Finally, according to (i) of Lemma 3.3, completing the proof of the theorem.
It is natural to ask under which condition * * is a homomorphism from Id * R onto Ann * R. A sufficient condition is presented by the following theorem. for every family (I s ; s ∈ S) of ideals of R. The rest of the proof is clear.
For any commutative semiring R and any ∈ Con R we call [0] the kernel of .
In contrast to the case of commutative rings, an ideal of a commutative semiring R need not be the kernel of some congruence on R. We have only [0] ∈ Id R provided ∈ Con R. A weaker result holds for the so-called Bourne congruence induced by an ideal I of R: Theorem 3.12 Let I be some ideal of a commutative semiring R = (R, +, ·, 0) and put

The (I ) ∈ Con R and I ⊆ [0] (I ).
The proof is easy (cf. Proposition 6.50 in Golan 1999) and therefore omitted.

Ideals and congruence kernels in Łukasiewicz semirings
In the following we investigate commutative semirings with a unary operation . For such extended semirings we study complements in the lattice of ideals which are not derived by annihilators as above, but by means of this unary operation. The concept of a Łukasiewicz semiring was introduced in Bonzio et al. (2016) and Chajda et al. (2018) as an algebraic semantic of a certain non-classical logic used in quantum mechanics. The motivation can be found in Chajda et al. (2018) and Bonzio et al. (2016).
Namely, within the logic of quantum mechanics effects are described by means of the so-called effect algebras. Z. Riečanová proved that every lattice effect algebra is built up by blocks which are in fact MV-algebras. MV-algebras were introduced in the 1950's by C. C. Chang as an algebraic axiomatization of Łukasiewicz many-valued logics. As shown in Chajda et al. (2018), MV-algebras are in oneto-one correspondence with Łukasiewicz semirings. Using these semirings instead of MV-algebras, one can apply the theory of semirings developed, e.g. in Golan (1999), i.e. the theory of ideals and annihilators in particular.
The special case y = 0 in (iii) yields x x ≈ 0. As shown in Chajda et al. (2018), the following identity holds in every Łukasiewicz semiring: x + y ≈ ((x y ) y ) .
If a ∈ R satisfies a 2 = a, then the previous identity yields a + a = ((aa) a) = (a a) = 0 = 1.
Since Łukasiewicz semirings are commutative semirings, we use the same definition of an ideal as before.
Lemma 4.1 Let R = (R, +, ·, , 0, 1) be a Łukasiewicz semiring and I a subset of R with 0 ∈ I satisfying If x ∈ R and y, x y ∈ I , then x ∈ I .
Then I ∈ Id R.
Proof Let a, b ∈ I and c ∈ R. We have (a b)b = a bb = 0 ∈ I whence a b ∈ I according to (5). Moreover, (a + b)a = aa + ba = a b ∈ I whence a + b ∈ I again according to (5). Finally, (ac)a = aa c = 0 ∈ I whence ac ∈ I according to (5) completing the proof of the lemma.
For every Łukasiewicz semiring R = (R, +, ·, , 0, 1) let CK(R) denote the set of all subsets I of R with 0 ∈ I satisfying (5) and the following condition: x, y, z ∈ R and x y ∈ I imply xz(yz) ∈ I .
According to Lemma 4.1, CK(R) ⊆ Id R. Put CK R := (CK R, ⊆). Next we prove that CK R is just the set of congruence kernels of R. Theorem 4.3 Let R = (R, +, ·, , 0, 1) be a Łukasiewicz semiring and I ∈ CK R. Put Then (I ) ∈ Con R.
Proof Let a, b, c ∈ R. It is evident that (I ) is reflexive and symmetric. Now assume (a, b), (b, c) we have ac , a c ∈ I according to (5) Theorem 4.4 Let R = (R, +, ·, , 0, 1) be a Łukasiewicz semiring. Then CK R is isomorphic to Con R := (Con R, ⊆) and hence a complete lattice. The infimum in CK R coincides with the set-theoretical intersection. Moreover, the correspondence described in Lemma 4.2 and Theorem 4.3 is one-to-one. = 0 a (b a) a = (a b) b 0  Since in a Łukasiewicz semiring R = (R, +, ·, , 0, 1) we have x ∨ y = x + y and since is an antitone involution on the poset (R, ≤) corresponding to the join-semilattice (R, +), we can use De Morgan laws and obtain x ∧ y = (x + y ) .
The following lemma lists some important properties of Łukasiewicz semirings.
(v) This follows since R is unitary.
For any two ideals I , J of a commutative semiring R = (R, +, ·, 0) we have I J ⊆ I ∩ J , but in general we do not have equality. However, in some cases equality follows.

Direct product of a commutative semiring and a finite unitary Boolean ring
If R 1 = (R 1 , +, ·, 0) and R 2 = (R 2 , +, ·, 0) are commutative semirings, I 1 ∈ Id R 1 and I 2 ∈ Id R 2 then I 1 × I 2 ∈ Id(R 1 × R 2 ). An ideal of R 1 × R 2 which is not of this form is called skew. If R 1 ×R 2 has no skew ideals, then Id(R 1 ×R 2 ) ∼ = Id R 1 × Id R 2 and hence Id(R 1 × R 2 ) is complemented if and only if Id R 1 and Id R 2 have this property. We are going to show that if one of two given commutative semirings is a finite unitary Boolean ring then the lattice of ideals of their direct product turns out to be directly decomposable. Recall that a ring is called Boolean if it satisfies the identity x 2 ≈ x. It is well known that such rings are commutative and satisfy the identity x + x ≈ 0.
Corollary 5.2 If R is a commutative semiring whose lattice of ideals is complemented or modular or distributive and B is a finite unitary Boolean ring, then R × B is a commutative semiring whose lattice of ideals is complemented or modular or distributive, respectively.
Hence, if R denotes the commutative semiring from Example 2.2 whose lattice of ideals is complemented (but complements do not coincide with annihilators in general) and B is a finite unitary Boolean ring, then R × B is a commutative semiring whose lattice of ideals is complemented, too. Hence, using our procedure, we may produce infinitely many commutative semirings having the mentioned property.
We are going to show that the finite unitary Boolean ring in the previous corollary cannot be substituted by a bounded distributive lattice even in the case when this finite unitary Boolean ring is the two-element one.

Example 5.3
Consider the Kleinian four-element group as the additive group of a zero-ring. Denote this ring by K. Now consider the two-element unitary Boolean ring B and put R := K × B. Since R is a ring, its lattice of ideals is modular (even complemented) and its Hasse diagram is shown in Fig. 4. The structure of Id R follows directly from Lemma 5.1 since Id R = Id(K × B 2 ) ∼ = Id K × Id B 2 ∼ = M 3 × 2.
If, however, we substitute B by the two-element distributive lattice (considered as a commutative semiring), we obtain a semiring R 1 which is not a ring and whose lattice of ideals contains skew ideals. This lattice of ideals is not modular. Its Hasse diagram is depicted in Fig. 5.
However, Id R 1 is complemented: {0} is a complement of R, b is a complement of i, j, n and o, c is a complement of h, d is a complement of m, and l is a complement of a, e, f , g and k.