Semigroup varieties of inflations of unions of groups

Clarke, G. T.

doi:10.1007/bf02676655

Cited by 12 publications

(8 citation statements)

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“…If F (ϕ) = {·}, then there are repetitions of variables in ϕ and applying Lemma 3.3 we get the statement. Now we prove (2). It must be Cp (ϕ) 3.…”

Section: Lemma 32 Let ϕ Be a Term Of Type τ 0 Such That There Existmentioning

confidence: 96%

“…Firstly, the binary part of Theorem 5.5 was proved in J. P lonka [11], Example 3, however we recall here some considerations from [11] for the convenience of the reader. Later, groupoids satisfying (2.ii) and xy ≈ x 2 y 2 were characterized as inflations of an idempotent semigroup in G. T. Clarke [2]. In this paper we characterize algebras from the varieties V k for k ∈ {1, 2, 3} by means of the P lonka sum construction.…”

Section: Representation Theoremmentioning

confidence: 98%

“…(3): Since a homomorphic image of an algebra is an algebra, so (2) implies that the set I is closed under · and f A . It follows also from (2) that I is idempotent, i.e.…”

Section: Unarily Extended Semilattices 109mentioning

confidence: 99%

“…We hope that the structure of algebras from the varieties V k for k ∈ {1, 2, 3} is clear for the reader, since f yields a retraction in algebras from V 1 , f yields a permutation of order at most 2 in algebras from V 2 and f yields a mapping of order at most 2, i.e., f [3] (x) ≈ f [2] (x) ≈ x 2 , in algebras from V 3 .…”

Section: Unarily Extended Semilattices 109mentioning

confidence: 99%

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Unarily Extended Semilattices

Marczak

Płonka

2008

Asian-European J. Math.

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In this paper we give equational and structural characterizations of all algebras representing the sequence (0, 3, 1, 1, . . . ). In the first part of the paper we prove that an algebra without constants has exactly three distinct essentially unary term operations and exactly one essentially n-ary term operation for every n > 1 if and only if it is clone equivalent to a generic of one of three varieties of algebras defined in Section 2. The second part of the paper is devoted to the representations of algebras from these three varieties by means of semilattice ordered systems of algebras.

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“…If F (ϕ) = {·}, then there are repetitions of variables in ϕ and applying Lemma 3.3 we get the statement. Now we prove (2). It must be Cp (ϕ) 3.…”

Section: Lemma 32 Let ϕ Be a Term Of Type τ 0 Such That There Existmentioning

confidence: 96%

Section: Representation Theoremmentioning

confidence: 98%

“…(3): Since a homomorphic image of an algebra is an algebra, so (2) implies that the set I is closed under · and f A . It follows also from (2) that I is idempotent, i.e.…”

Section: Unarily Extended Semilattices 109mentioning

confidence: 99%

Section: Unarily Extended Semilattices 109mentioning

confidence: 99%

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Unarily Extended Semilattices

Marczak

Płonka

2008

Asian-European J. Math.

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“…The union of these new sets then forms the base set of a new algebra, in which operations are performed by the rule that any element in the set C a always acts like a. For more information on the inflation construction, see [4]. Now let A = (A; ∨, ∧) be a lattice.…”

Section: The 3-level Inflation Constructionmentioning

confidence: 99%

2-Normalization of lattices

2008

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Let τ be a type of algebras. A valuation of terms of type τ is a function v assigning to each term t of type τ a value v(t) 0. For k 1, an identity s ≈ t of type τ is said to be k-normal (with respect to valuation v) if either s = t or both s and t have value k. Taking k = 1 with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called k-normal (with respect to the valuation v) if all its identities are k-normal. For any variety V , there is a least k-normal variety N k (V ) containing V , namely the variety determined by the set of all k-normal identities of V . The concept of k-normalization was introduced by K. Denecke and S. L. Wismath in their paper (Algebra Univers., 50, 2003, pp.107-128) and an algebraic characterization of the elements of N k (V ) in terms of the algebras in V was given in (Algebra Univers., 51, 2004, pp. 395-409). In this paper we study the algebras of the variety N 2 (V ) where V is the type (2, 2) variety L of lattices and our valuation is the usual depth valuation of terms. We introduce a construction called the 3-level inflation of a lattice, and use the order-theoretic properties of lattices to show that the variety N 2 (L) is precisely the class of all 3-level inflations of lattices. We also produce a finite equational basis for the variety N 2 (L).

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Decompositions of semigroups induced by identities

Ćirić

Bogdanović

1993

Semigroup Forum

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Semigroup varieties of inflations of unions of groups

Cited by 12 publications

References 3 publications

Unarily Extended Semilattices

Unarily Extended Semilattices

2-Normalization of lattices

Decompositions of semigroups induced by identities

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