On surjective kernels of partial algebras.
The paper deals with the characterization of ordered sets by means of ternary semigroups of homomorphisms of ordered sets.
We prove that every clone of operations on a finite set , if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting for some finitary relation over . It follows that for a fixed finite set , the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few...
he class of all M-solid varieties of a given type t forms a complete sublattice of the lattice ℒ(τ) of all varieties of algebrasof type t. This gives a tool for a better description of the lattice ℒ(τ) by characterization of complete sublattices. In particular, this was done for varieties of semigroups by L. Polák ([10]) as well as by Denecke and Koppitz ([4], [5]). Denecke and Hounnon characterized M-solid varieties of semirings ([3]) and M-solid varieties of groups were characterized by Koppitz...
Abstract characterizations of relations of nonempty intersection, inclusion end equality of domains for partial -place functions are presented. Representations of Menger -semigroups by partial -place functions closed with respect to these relations are investigated.
Irregular (quasi)varieties of groupoids are (quasi)varieties that do not contain semilattices. The regularization of a (strongly) irregular variety of groupoids is the smallest variety containing and the variety of semilattices. Its quasiregularization is the smallest quasivariety containing and . 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...
In this paper the notion of a ternary semigroup of morphisms of objects in a category is introduced. The connection between an isomorphism of categories and an isomorphism of ternary semigroups of morphisms of suitable objects in these categories is considered. Finally, the results obtained for general categories are applied to the categories and which were studied in [5].
A method is presented for proving primality and functional completeness theorems, which makes use of the operation-relation duality. By the result of Sierpiński, we have to investigate relations generated by the two-element subsets of only. We show how the method applies for proving Słupecki’s classical theorem by generating diagonal relations from each pair of k-tuples.
The aim of this paper is to describe how varieties of algebras of type τ can be classified by using the form of the terms which build the (defining) identities of the variety. There are several possibilities to do so. In [3], [19], [15] normal identities were considered, i.e. identities which have the form x ≈ x or s ≈ t, where s and t contain at least one operation symbol. This was generalized in [14] to k-normal identities and in [4] to P-compatible identities. More generally, we select a subset...
A characterization of all classes of idempotent groupoids having no more than two essentially binary term operations with respect to small finite models is given.