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Selfdistributive groupoids of small orders

Jaroslav Ježek, Tomáš Kepka (1997)

Czechoslovak Mathematical Journal

After enumerating isomorphism types of at most five-element left distributive groupoids, we prove that a distributive groupoid with less than 81 elements is necessarily medial.

Simple balanced groupoids

Tomáš Kepka, Petr Němec (1996)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Simple zeropotent paramedial groupoids are balanced

Robert El Bashir, Jaroslav Ježek, Tomáš Kepka (2000)

Czechoslovak Mathematical Journal

This short note is a continuation of and and its purpose is to show that every simple zeropotent paramedial groupoid containing at least three elements is strongly balanced in the sense of .

Slim groupoids

Jaroslav Ježek (2007)

Czechoslovak Mathematical Journal

Slim groupoids are groupoids satisfying x ( y z ) x ̄ z . We find all simple slim groupoids and all minimal varieties of slim groupoids. Every slim groupoid can be embedded into a subdirectly irreducible slim groupoid. The variety of slim groupoids has the finite embeddability property, so that the word problem is solvable. We introduce the notion of a strongly nonfinitely based slim groupoid (such groupoids are inherently nonfinitely based) and find all strongly nonfinitely based slim groupoids with at most four...

Small idempotent clones. I

Józef Dudek (1998)

Czechoslovak Mathematical Journal

G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning p n -sequences and free spectra of algebras to the topic “Small idempotent clones” (see Section 6 of [18]). Many authors, e.g., [8], [14, 15], [22], [25] and [29, 30] were interested in p n -sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids ( G , · ) with p 2 ( G , · ) ≤ 2 (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26,...

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...

Some results in bipolar-valued fuzzy ordered AG-groupoids

Faisal, Naveed Yaqoob, Arsham Borumand Saeid (2012)

Discussiones Mathematicae - General Algebra and Applications

In this paper, we introduce the concept of bipolar-valued fuzzification of ordered 𝓐𝓖-groupoids and discuss some structural properties of bipolar-valued fuzzy two-sided ideals of an intra-regular ordered 𝓐𝓖-groupoid.

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.

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