Varieties of Commutative Semigroups

Kisielewicz, Andrzej

doi:10.2307/2154694

Cited by 9 publications

(14 citation statements)

References 16 publications

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“…REMARK 3.2. The above theorem does not hold for infinite semigroups, and the paper [20] provides a counterexample. It suffices to take the semigroup generating the variety defined by identities xy = yx and x 3 y 2 = x 2 y 2 , for it has a polynomially bounded p nsequence, but fails to satisfy any of the conditions of the above theorem.…”

Section: The Main Resultsmentioning

confidence: 98%

Finite semigroups with slowly growing p n -sequences

Crvenković¹,

Dolinka²

2000

Algebra Universalis

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The p n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p n (S) ≤ cn r holds for all n ≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p nsequence and give some examples.

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Section: The Main Resultsmentioning

confidence: 98%

Finite semigroups with slowly growing p n -sequences

Crvenković¹,

Dolinka²

2000

Algebra Universalis

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“…The set of equational theories of permutative semigroups forms a lattice that contains the lattice of equational theories of commutative semigroups. An extension of Kisielewicz's [13] description to the lattice of all equational theories of permutative semigroups does not seem easy. The main obstacle is that, as it was proved in [18], there are equational theories of permutative semigroups that fail to be finitely based.…”

Section: Mariusz Grechmentioning

confidence: 99%

“…One of the reasons that the problems above seem very hard is that we have very little detailed knowledge about the structure of the lattice of equational theories of semigroups. In contrast, we have very good knowledge about the structure of the lattice L(Com) of equational theories of commutative semigroups ( [13] and [7,4]). Using this, in [14], A. Kisielewicz has proved that many sets and individual theories in L(Com) are definable.…”

Section: Introductionmentioning

confidence: 99%

The structure and definability in the lattice of equational theories of strongly permutative semigroups

Grech

2012

Trans. Amer. Math. Soc.

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In this paper, we study the structure and the first-order definability in the lattice L(SP) of equational theories of strongly permutative semigroups, that is, semigroups satisfying a permutation identity x 1 • • • x n = x σ(1) • • • x σ(n) with σ(1) > 1 and σ(n) < n. We show that each equational theory of such semigroups is described by five objects: an order filter, an equivalence relation, and three integers. We fully describe the lattice L(SP); inclusion, operations ∨ and ∧, and covering relation. Using this description, we prove, in particular, that each individual theory of strongly permutative semigroups is definable, up to duality.

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“…With this characterization at hand, it is possible to obtain more information on commutative varieties with polynomially bounded p n -sequences and their position in the lattice of semigroup varieties using the results of Kisielewicz [14].…”

Section: Dolinkamentioning

confidence: 99%