The Lattice of Equational Classes of Algebras with One Unary Operation

Jacobs, Eugene A.; Schwabauer, Robert

doi:10.2307/2311743

Cited by 12 publications

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“…(i) The fact that L has exactly two atoms was already remarked in [3]. In that paper it was also proved how to calculate the join and the meet of two elements of L.…”

Section: èöóó the Above Description Of L Yieldsmentioning

confidence: 87%

A note on normal varieties of monounary algebras

Chajda¹,

Länger²

2002

Czechoslovak Mathematical Journal

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“…(i) The fact that L has exactly two atoms was already remarked in [3]. In that paper it was also proved how to calculate the join and the meet of two elements of L.…”

Section: èöóó the Above Description Of L Yieldsmentioning

confidence: 87%

A note on normal varieties of monounary algebras

Chajda¹,

Länger²

2002

Czechoslovak Mathematical Journal

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“…In type (1), with a single unary operation symbol f , we use the notation f 0 (x) = x and inductively f k (x) = f (f (· · · f (x) · · ·)). The structure of the varieties and identities of this type are well known; see [9] for details. We consider the variety V = Mod {x ≈ f n (x)} of type τ = (1), noting that an identity f k (x) ≈ f l (x) ∈ Id V iff k ≡ l mod n. It was proved in [13] that V is solid.…”

Section: Proposition 32 For Any Integer N ≥ 1 There Is a Variety V mentioning

confidence: 99%

The Dimension of a Variety and the Kernel of a Hypersubstitution

Denecke

Wismath

2009

Int. J. Algebra Comput.

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The dimension of a variety V of algebras of a given type was introduced by E. Graczyńska and D. Schweigert in [7] as the cardinality of the set of all derived varieties of V which are properly contained in V . In this paper, we characterize all solid varieties of dimensions 0, 1, and 2; prove that the dimension of a variety of finite type is at most ℵ 0 ; give an example of a variety which has infinite dimension; and show that for every n ∈ N there is a variety with dimension n. Finally, we show that the dimension of a variety is related to the concept of the semantical kernel of a hypersubstitution and apply this connection to calculate the dimension of the class of all algebras of type τ = (n).

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“…P r o o f. According to Gerhard [2], all varieties of idempotent semigroups are finitely based. According to Jacobs and Schwabauer [3], all varieties of algebras with one unary operation are finitely based. Thus it remains to consider the at most three-element idempotent slim groupoids that are not semigroups and do not satisfy xy ≈ xz.…”

Section: I-strongly Nonfinitely Based Slim Groupoidsmentioning

confidence: 99%

Varieties of idempotent slim groupoids

Ježek

2007

Czech Math J

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The Lattice of Equational Classes of Algebras with One Unary Operation

Cited by 12 publications

References 0 publications

A note on normal varieties of monounary algebras

A note on normal varieties of monounary algebras

The Dimension of a Variety and the Kernel of a Hypersubstitution

Varieties of idempotent slim groupoids

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