Rectangular groupoids
Regular elements and Green's relations in Menger algebras of terms
Defining an (n+1)-ary superposition operation on the set of all n-ary terms of type τ, one obtains an algebra of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation there are different possibilities to define binary associative operations on the set and on the cartesian power . In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative operations...
Relational classes and their characterizations
Remarks on -independence of Stone algebras
Retracts and Q-independence
A non-empty set X of a carrier A of an algebra A is called Q-independent if the equality of two term functions f and g of the algebra A on any finite system of elements a₁,a₂,...,aₙ of X implies f(p(a₁),p(a₂),...,p(aₙ)) = g(p(a₁),p(a₂),...,p(aₙ)) for any mapping p ∈ Q. An algebra B is a retract of A if B is the image of a retraction (i.e. of an idempotent endomorphism of B). We investigate Q-independent subsets of algebras which have a retraction in their set of term functions.
Semi-primal clusters
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 properties of residuated lattices
We investigate some (universal algebraic) properties of residuated lattices—algebras which play the role of structures of truth values of various systems of fuzzy logic.
Some remarks on distributive groupoids
Strictly order primal algebras.
Structure of free algebras
Structures Algebraiques munis, avec des lois de composition generalisees - Generalisations du produit scalaire
Structures naturelles des demi-groupes et des anneaux réguliers ou involutés
Certaines relations binaires sont définies sur les demi-groupes et les demi-groupes à involution. On examine comment elles peuvent en ordonner les éléments: notamment les idempotents, les éléments réguliers au sens de von Neumann, ceux qui possédent un inverse ponctuel ou de Moore-Penrose ; et en fonction aussi de conditions sur l'involution. Ces relations peuvent alors coïncider avec les ordres naturels des idempotents et des demi-groupes inverses, avec les ordres de Drazin et de Hartwig : elles...
Sur le rang et la définissabilité des opérations implicites dans les classes d'algèbres
TC semigroups and inflations.
Term functions on non-Abelian groups of order
Terms and semiterms
The dimension of a variety
Derived varieties were invented by P. Cohn in [4]. Derived varieties of a given type were invented by the authors in [10]. In the paper we deal with the derived variety of a given variety, by a fixed hypersubstitution σ. We introduce the notion of the dimension of a variety as the cardinality κ of the set of all proper derived varieties of V included in V. We examine dimensions of some varieties in the lattice of all varieties of a given type τ. Dimensions of varieties of lattices and all subvarieties...
The Galois correspondence between subvariety lattices and monoids of hpersubstitutions
Denecke and Reichel have described a method of studying the lattice of all varieties of a given type by using monoids of hypersubstitutions. In this paper we develop a Galois correspondence between monoids of hypersubstitutions of a given type and lattices of subvarieties of a given variety of that type. We then apply the results obtained to the lattice of varieties of bands (idempotent semigroups), and study the complete sublattices of this lattice obtained through the Galois correspondence.
The minimal extension of sequences III. On problem 16 of Grätzer and Kisielewicz
The main result of this paper is a description of totally commutative idempotent groupoids. In particular, we show that if an idempotent groupoid (G,·) has precisely m ≥ 2 distinct essentially binary polynomials and they are all commutative, then G contains a subgroupoid isomorphic to the groupoid described below. In [2], this fact was proved for m = 2.