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Some regular quasivarieties of commutative binary modes

K. Matczak, Anna B. Romanowska (2014)

Commentationes Mathematicae Universitatis Carolinae

Irregular (quasi)varieties of groupoids are (quasi)varieties that do not contain semilattices. The regularization of a (strongly) irregular variety 𝒱 of groupoids is the smallest variety containing 𝒱 and the variety 𝒮 of semilattices. Its quasiregularization is the smallest quasivariety containing 𝒱 and 𝒮 . In an earlier paper the authors described the lattice of quasivarieties of cancellative commutative binary modes, i.e. idempotent commutative and entropic (or medial) groupoids. They are all irregular...

Split extensions and semidirect products of unitary magmas

Marino Gran, George Janelidze, Manuela Sobral (2019)

Commentationes Mathematicae Universitatis Carolinae

We develop a theory of split extensions of unitary magmas, which includes defining such extensions and describing them via suitably defined semidirect product, yielding an equivalence between the categories of split extensions and of (suitably defined) actions of unitary magmas on unitary magmas. The class of split extensions is pullback stable but not closed under composition. We introduce two subclasses of it that have both of these properties.

The globals of pseudovarieties of ordered semigroups containing B 2 and an application to a problem proposed by Pin

Jorge Almeida, Ana P. Escada (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Given a basis of pseudoidentities for a pseudovariety of ordered semigroups containing the 5-element aperiodic Brandt semigroup B 2 , under the natural order, it is shown that the same basis, over the most general graph over which it can be read, defines the global. This is used to show that the global of the pseudovariety of level 3 / 2 of Straubing-Thérien’s concatenation hierarchy has infinite vertex rank.

The globals of pseudovarieties of ordered semigroups containing B2 and an application to a problem proposed by Pin

Jorge Almeida, Ana P. Escada (2010)

RAIRO - Theoretical Informatics and Applications

Given a basis of pseudoidentities for a pseudovariety of ordered semigroups containing the 5-element aperiodic Brandt semigroup B2, under the natural order, it is shown that the same basis, over the most general graph over which it can be read, defines the global. This is used to show that the global of the pseudovariety of level 3/2 of Straubing-Thérien's concatenation hierarchy has infinite vertex rank.

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