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Regular elements and Green's relations in Menger algebras of terms

Klaus Denecke, Prakit Jampachon (2006)

Discussiones Mathematicae - General Algebra and Applications

Defining an (n+1)-ary superposition operation S n on the set W τ ( X n ) of all n-ary terms of type τ, one obtains an algebra n - c l o n e τ : = ( W τ ( X n ) ; S n , x 1 , . . . , x n ) of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation S n there are different possibilities to define binary associative operations on the set W τ ( X n ) and on the cartesian power W τ ( X n ) n . In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative operations...

Retracts and Q-independence

Anna Chwastyk (2007)

Discussiones Mathematicae - General Algebra and Applications

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

D. James Samuelson (1970)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Small idempotent clones. I

Józef Dudek (1998)

Czechoslovak Mathematical Journal

G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning p n -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 p n -sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids ( G , · ) with p 2 ( G , · ) ≤ 2 (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26,...

Some properties of residuated lattices

Radim Bělohlávek (2003)

Czechoslovak Mathematical Journal

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.

Structures naturelles des demi-groupes et des anneaux réguliers ou involutés

Jean Calmes (1994)

Mathématiques et Sciences Humaines

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...

The dimension of a variety

Ewa Graczyńska, Dietmar Schweigert (2007)

Discussiones Mathematicae - General Algebra and Applications

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 V σ 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

Klaus Denecke, Jennifer Hyndman, Shelly L. Wismath (2000)

Discussiones Mathematicae - General Algebra and Applications

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

J. Dudek (1996)

Colloquium Mathematicae

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 N m described below. In [2], this fact was proved for m = 2.

The monoid of generalized hypersubstitutions of type τ = (n)

Wattapong Puninagool, Sorasak Leeratanavalee (2010)

Discussiones Mathematicae - General Algebra and Applications

A (usual) hypersubstitution of type τ is a function which takes each operation symbol of the type to a term of the type, of the same arity. The set of all hypersubstitutions of a fixed type τ forms a monoid under composition, and semigroup properties of this monoid have been studied by a number of authors. In particular, idempotent and regular elements, and the Green’s relations, have been studied for type (n) by S.L. Wismath. A generalized hypersubstitution of type τ=(n) is a mapping σ which takes...

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