On Grätzer's problem of binary 1-step congruence schemes
On idempotent modifications of -algebras
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 -algebra we denote by and the idempotent modification, the underlying set or the underlying lattice of , respectively. In the present paper we prove that if is semisimple and is a chain, then is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras.
On Jónsson's theorem
A proof of Jonsson's theorem inspired by considering a natural topology on algebraic lattices is given.
On lattices of varieties of universal algebras
On locally small based algebraic theories
On modular elements of the lattice of semigroup varieties
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 is called substitutive if the words and depend on the same letters and may be obtained from by renaming of letters.) We completely determine all commutative modular...
On modularity in lattices of congruences on ordered sets
On multiplication groups of relatively free quasigroups isotopic to Abelian groups
If is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group is a Frobenius group. Conversely, if is a Frobenius group, a quasigroup, then has to be isotopic to an Abelian group. If is, in addition, finite, then it must be a central quasigroup (a -quasigroup).
On -permutable and distributive at 0 varieties.
On nondistributive Steiner quasigroups
A well known result of R. Dedekind states that a lattice is nonmodular if and only if it has a sublattice isomorphic to . Similarly a lattice is nondistributive if and only if it has a sublattice isomorphic to or (see [11]). Recently a few results in this spirit were obtained involving the number of polynomials of an algebra (see e.g. [1], [3], [5], [6]). In this paper we prove that a nondistributive Steiner quasigroup (G,·) has at least 21 essentially ternary polynomials (which improves the...
On n-permutable equivalence relations in regular categories
On one problem in the theory of partial monounary algebras
On permutability in semigroup varieties
The paper contains characterizations of semigroup varieties whose semigroups with one generator (two generators) are permutable. Here all varieties of regular -semigroups are described in which each semigroup with two generators is permutable.
On primary and reduced varieties of monoassociative algebras.
On quasivarieties of nilpotent Moufang loops. I
In this part the smallest non-abelian quasivarieties for nilpotent Moufang loops are described.
On quasivarieties of nilpotent Moufang loops. II
In this part of the paper we study the quasiidentities of the nilpotent Moufang loops. In particular, we solve the problem of finite basis for quasiidentities in the finitely generated nilpotent Moufang loop.
On quasivarieties of partial algebras, their generation and their subquasivarieties. I
On radical classes of Abelian linearly ordered groups
On reductive and distributive algebras
The paper investigates idempotent, reductive, and distributive groupoids, and more generally -algebras of any type including the structure of such groupoids as reducts. In particular, any such algebra can be built up from algebras with a left zero groupoid operation. It is also shown that any two varieties of left -step reductive -algebras, and of right -step reductive -algebras, are independent for any positive integers and . This gives a structural description of algebras in the join of...
On schemes for congruence distributivity
We present diagrammatic schemes characterizing congruence 3-permutable and distributive algebras. We show that a congruence 3-permutable algebra is congruence meetsemidistributive if and only if it is distributive. We characterize varieties of algebras satisfying the so-called triangular scheme by means of a Maltsev-type condition.