Minimal varieties of semirings

Polin, S. V.

doi:10.1007/bf01140525

Cited by 5 publications

(3 citation statements)

References 2 publications

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“…In the literature of semirings there are several definitions depending on whether the algebra contains an identity and/or a zero element. Polin [ 10 ] studied minimal varieties of semirings without 0, 1 as constant operations. A variety is minimal if it has no proper subvarieties other than the variety of one-element algebras.…”

Section: Introductionmentioning

confidence: 99%

Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests

Alpay

Jipsen

2020

Relational and Algebraic Methods in Computer Science

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A commutative doubly-idempotent semiring (cdi-semiring) is a semilattice with and a semilattices with identity 1 such that , and holds for all . Bounded distributive lattices are cdi-semirings that satisfy , and the variety of cdi-semirings covers the variety of bounded distributive lattices. Chajda and Länger showed in 2017 that the variety of all cdi-semirings is generated by a 3-element cdi-semiring. We show that there are seven cdi-semirings with a -semilattice of height less than or equal to 2. We construct all cdi-semirings for which their multiplicative semilattice is a chain with elements, and we show that up to isomorphism the number of such algebras is the Catalan number . We also show that cdi-semirings with a complete atomic Boolean -semilattice on the set of atoms A are determined by singleton-rooted preorder forests on the set A . From these results we obtain efficient algorithms to construct all multiplicatively linear cdi-semirings of size n and all Boolean cdi-semirings of size .

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Section: Introductionmentioning

confidence: 99%

Commutative Doubly-Idempotent Semirings Determined by Chains and by Preorder Forests

Alpay

Jipsen

2020

Relational and Algebraic Methods in Computer Science

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“…First we recall a notion from [4]. For an (involution) semiring A and a, b ∈ A we define a relation by:…”

Section: Lemma 4 Let a Be A Semiring With Involution Which Is Not Admentioning

confidence: 99%

“…Then the relation θ defined by aθb if and only if a b and b a is (by Lemma 3.1 of [4]) a congruence of A (it is immediate to verify that θ is compatible with the involution operation) and, moreover, the quotient A/θ is additively idempotent, yielding that θ must be the universal relation on A. In other words, for every element a ∈ A we have eθa, which is the same as saying that there exists a natural number n ≥ 2 such that na = e. Vol.…”

Section: Lemma 4 Let a Be A Semiring With Involution Which Is Not Admentioning

confidence: 99%