Pointwise and global sums and negatives of binary relations.
Pointwise and global sums and negatives of translation relations.
Powers and alternative laws
A groupoid is alternative if it satisfies the alternative laws and . These laws induce four partial maps on
Pseudo-complemented distributive groupoids
Quasigroups of fractions
Quasitrivial left distributive groupoids
Left distributive quasitrivial groupoids are completely described and those of them which are subdirectly irreducible are found. There are also determined all left distributive algebras such that is a quasitrivial groupoid.
Quotients and homomorphisms of relational systems
Relational systems containing one binary relation are investigated. Quotient relational systems are introduced and some of their properties are characterized. Moreover, homomorphisms, strong mappings and cone preserving mappings are introduced and the interplay between these notions is considered. Finally, the connection between directed relational systems and corresponding groupoids is investigated.
Representation of undirected graphs by anticommutative conservative groupoids
The paper studies tolerances and congruences on anticommutative conservative groupoids. These groupoids can be assigned in a one-to-one way to undirected graphs.
Retracts of monounary algebras corresponding to groupoids
Selfdistributive groupoids of small orders
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.
Selfdistributive groupoids. Part A1. Non-indempotent left distributive groupoids
Selfdistributive groupoids. Part D1: Left distributive semigroups
Selfdistributive grupoids, Part A2: Non-idempotent left distributive quasigroups
Simple balanced groupoids
Simple paramedial groupoids
Simple zeropotent paramedial groupoids are balanced
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
Slim groupoids are groupoids satisfying . 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
G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning -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 -sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids with (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26,...
Some regular quasivarieties of commutative binary modes
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...
Some results in bipolar-valued fuzzy ordered AG-groupoids
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.