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

100%

Denecke K., Jampachon P.

Discussiones Mathematicae - General Algebra and Applications

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2006

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tom 26

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nr 1

85-109

EN

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 $n-clone τ := (W_{τ}(X_n); S^n, x_1, ..., x_n)$ 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 as fundamental operations.

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T-Varieties and Clones of T-terms

100%

Denecke K., Jampachon P.

Discussiones Mathematicae - General Algebra and Applications

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2005

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tom 25

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nr 1

89-101

EN

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 T of $W_{τ}(X)$, the set of all terms of type τ, and consider identities from T×T. Since any variety can be described by one heterogenous algebra, its clone, we are also interested in the corresponding clone-like structure. Identities of the clone of a variety V correspond to M-hyperidentities for certain monoids M of hypersubstitutions. Therefore we will also investigate these monoids and the corresponding M-hyperidentities.

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Non-Deterministic Linear Hypersubstitutions

100%

Lekkoksung N., Jampachon P.

Discussiones Mathematicae - General Algebra and Applications

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2015

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tom 35

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nr 1

97-103

EN

A non-deterministic hypersubstitution maps operation symbols to sets of terms of the corresponding arity. A non-deterministic hypersubstitution of type τ is said to be linear if it maps any operation symbol to a set of linear terms of the corresponding arity. We show that the extension of non-deterministic linear hypersubstitutions of type τ map sets of linear terms to sets of linear terms. As a consequence, the collection of all non-deterministic linear hypersubstitutions forms a monoid. Non-deterministic linear hypersubstitutions can be applied to identities and to algebras of type τ.

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Four-part semigroups - semigroups of Boolean operations

81%

Jampachon P., Susanti Y., Denecke K.

Discussiones Mathematicae - General Algebra and Applications

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2012

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tom 32

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nr 1

115-136

EN

Four-part semigroups form a new class of semigroups which became important when sets of Boolean operations which are closed under the binary superposition operation f + g := f(g,...,g), were studied. In this paper we describe the lattice of all subsemigroups of an arbitrary four-part semigroup, determine regular and idempotent elements, regular and idempotent subsemigroups, homomorphic images, Green's relations, and prove a representation theorem for four-part semigroups.

Ograniczanie wyników

4 Discussiones Mathematicae - General Algebra and Applications

4 Jampachon P.

3 Denecke K.

1 Lekkoksung N.

1 Susanti Y.

1 2015

1 2012

1 2006

1 2005

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

T-Varieties and Clones of T-terms

Non-Deterministic Linear Hypersubstitutions

Four-part semigroups - semigroups of Boolean operations