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On distributive trices

Kiyomitsu Horiuchi, Andreja Tepavčević (2001)

Discussiones Mathematicae - General Algebra and Applications

A triple-semilattice is an algebra with three binary operations, which is a semilattice in respect of each of them. A trice is a triple-semilattice, satisfying so called roundabout absorption laws. In this paper we investigate distributive trices. We prove that the only subdirectly irreducible distributive trices are the trivial one and a two element one. We also discuss finitely generated free distributive trices and prove that a free distributive trice with two generators has 18 elements.

On Equational Theory of Left Divisible Left Distributive Groupoids

Přemysl Jedlička (2012)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

It is an open question whether the variety generated by the left divisible left distributive groupoids coincides with the variety generated by the left distributive left quasigroups. In this paper we prove that every left divisible left distributive groupoid with the mapping a ↦ a 2 surjective lies in the variety generated by the left distributive left quasigroups.

On free modes

Michał Marek Stronkowski (2006)

Commentationes Mathematicae Universitatis Carolinae

We prove a theorem describing the equational theory of all modes of a fixed type. We use this result to show that a free mode with at least one basic operation of arity at least three, over a set of cardinality at least two, does not satisfy identities selected by ’A. Szendrei in Identities satisfied by convex linear forms, Algebra Universalis 12 (1981), 103–122, that hold in any subreduct of a semimodule over a commutative semiring. This gives a negative answer to the question raised by A. Romanowska:...

On free Turing algebras

Herbert Lugowski (1986)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

On idempotent modifications of M V -algebras

Ján Jakubík (2007)

Czechoslovak Mathematical Journal

The notion of idempotent modification of an algebra was introduced by Ježek. He proved that the idempotent modification of a group is subdirectly irreducible. For an M V -algebra 𝒜 we denote by 𝒜 ' , A and ℓ ( 𝒜 ) the idempotent modification, the underlying set or the underlying lattice of 𝒜 , respectively. In the present paper we prove that if 𝒜 is semisimple and ℓ ( 𝒜 ) is a chain, then 𝒜 ' is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras.

On Jónsson's theorem

Diego Vaggione (1996)

Mathematica Bohemica

A proof of Jonsson's theorem inspired by considering a natural topology on algebraic lattices is given.

On modular elements of the lattice of semigroup varieties

Boris M. Vernikov (2007)

Commentationes Mathematicae Universitatis Carolinae

A semigroup variety is called modular if it is a modular element of the lattice of all semigroup varieties. We obtain a strong necessary condition for a semigroup variety to be modular. In particular, we prove that every modular nil-variety may be given by 0-reduced identities and substitutive identities only. (An identity u = v is called substitutive if the words u and v depend on the same letters and v may be obtained from u by renaming of letters.) We completely determine all commutative modular...

On multiplication groups of relatively free quasigroups isotopic to Abelian groups

Aleš Drápal (2005)

Czechoslovak Mathematical Journal

If Q is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group M l t Q is a Frobenius group. Conversely, if M l t Q is a Frobenius group, Q a quasigroup, then Q has to be isotopic to an Abelian group. If Q is, in addition, finite, then it must be a central quasigroup (a T -quasigroup).

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