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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Hypersatisfaction of formulas in agebraic systems

100%
Denecke K., Phusanga D.
Discussiones Mathematicae - General Algebra and Applications
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2009
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tom 29
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nr 2
123-151
EN
In [2] the theory of hyperidentities and solid varieties was extended to algebraic systems and solid model classes of algebraic systems. The disadvantage of this approach is that it needs the concept of a formula system. In this paper we present a different approach which is based on the concept of a relational clone. The main result is a characterization of solid model classes of algebraic systems. The results will be applied to study the properties of the monoid of all hypersubstitutions of an ordered algebra.
2
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Nd-solid varieties

100%
Denecke K., Glubudom P.
Discussiones Mathematicae - General Algebra and Applications
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2007
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tom 27
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nr 2
245-262
EN
A non-deterministic hypersubstitution maps any operation symbol of a tree language of type τ to a set of trees of the same type, i.e. to a tree language. Non-deterministic hypersubstitutions can be extended to mappings which map tree languages to tree languages preserving the arities. We define the application of a non-deterministic hypersubstitution to an algebra of type τ and obtain a class of derived algebras. Non-deterministic hypersubstitutions can also be applied to equations of type τ. Formally, we obtain two closure operators which turn out to form a conjugate pair of completely additive closure operators. This allows us to use the theory of conjugate pairs of additive closure operators for a characterization of M-solid non-deterministic varieties of algebras. As an application we consider M-solid non-deterministic varieties of semigroups.
3
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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.
4
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The order of normalform hypersubstitutions of type (2)

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Denecke K., Mahdavi K.
Discussiones Mathematicae - General Algebra and Applications
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2000
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tom 20
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nr 2
183-192
EN
In [2] it was proved that all hypersubstitutions of type τ = (2) which are not idempotent and are different from the hypersubstitution whichmaps the binary operation symbol f to the binary term f(y,x) haveinfinite order. In this paper we consider the order of hypersubstitutionswithin given varieties of semigroups. For the theory of hypersubstitution see [3].
5
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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.
6
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Binary relations on the monoid of V-proper hypersubstitutions

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Denecke K., Srithus R.
Discussiones Mathematicae - General Algebra and Applications
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2006
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tom 26
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nr 2
233-251
EN
In this paper we consider different relations on the set P(V) of all proper hypersubstitutions with respect to a given variety V and their properties. Using these relations we introduce the cardinalities of the corresponding quotient sets as degrees and determine the properties of solid varieties having given degrees. Finally, for all varieties of bands we determine their degrees.
7
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Hyperidentities in many-sorted algebras

100%
Denecke K., Lekkoksung S.
Discussiones Mathematicae - General Algebra and Applications
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2009
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tom 29
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nr 1
47-74
EN
The theory of hyperidentities generalizes the equational theory of universal algebras and is applicable in several fields of science, especially in computers sciences (see e.g. [2,1]). The main tool to study hyperidentities is the concept of a hypersubstitution. Hypersubstitutions of many-sorted algebras were studied in [3]. On the basis of hypersubstitutions one defines a pair of closure operators which turns out to be a conjugate pair. The theory of conjugate pairs of additive closure operators can be applied to characterize solid varieties, i.e., varieties in which every identity is satisfied as a hyperidentity (see [4]). The aim of this paper is to apply the theory of conjugate pairs of additive closure operators to many-sorted algebras.
8
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Locally finite M-solid varieties of semigroups

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Denecke K., Pibaljommee B.
Discussiones Mathematicae - General Algebra and Applications
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2003
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tom 23
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nr 2
139-148
EN
An algebra of type τ is said to be locally finite if all its finitely generated subalgebras are finite. A class K of algebras of type τ is called locally finite if all its elements are locally finite. It is well-known (see [2]) that a variety of algebras of the same type τ is locally finite iff all its finitely generated free algebras are finite. A variety V is finitely based if it admits a finite basis of identities, i.e. if there is a finite set σ of identities such that V = ModΣ, the class of all algebras of type τ which satisfy all identities from Σ. Every variety which is generated by a finite algebra is locally finite. But there are finite algebras which are not finitely based. For semigroup varieties, Perkins proved that the variety generated by the five-element Brandt-semigroup $B¹₂ = { \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix}}$ is not finitely based ([9], [10]). An identity s ≈ t is called a hyperidentity of a variety V if whenever the operation symbols occurring in s and in t, respectively, are replaced by any terms of V of the appropriate arity, the identity which results, holds in V. A variety V is called solid if every identity of V also holds as a hyperidentity in V. If we apply only substitutions from a set M we speak of M-hyperidentities and M-solid varieties. In this paper we use the theory of M-solid varieties to prove that a type (2) M-solid variety of the form $V = H_{M}Mod{F(x₁,F(x₂,x₃)) ≈ F(F(x₁,x₂),x₃)}$, which consists precisely of all algebras which satisfy the associative law as an M -hyperidentity is locally finite iff the hypersubstitution which maps F to the word x₁x₂x₁ or to the word x₂x₁x₂ belongs to M and that V is finitely based if it is locally finite.
9
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Tree transformations defined by hypersubstitutions

100%
Arworn S., Denecke K.
Discussiones Mathematicae - General Algebra and Applications
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2001
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tom 21
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nr 2
219-227
EN
Tree transducers are systems which transform trees into trees just as automata transform strings into strings. They produce transformations, i.e. sets consisting of pairs of trees where the first components are trees belonging to a first language and the second components belong to a second language. In this paper we consider hypersubstitutions, i.e. mappings which map operation symbols of the first language into terms of the second one and tree transformations defined by such hypersubstitutions. We prove that the set of all tree transformations which are defined by hypersubstitutions of a given type forms a monoid with respect to the composition of binary relations which is isomorphic to the monoid of all hypersubstitutions of this type. We characterize transitivity, reflexivity and symmetry of tree transformations by properties of the corresponding hypersubstitutions. The results will be applied to languages built up by individual variables and one operation symbol of arity n ≥ 2.
10
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Complexity of hypersubstitutions and lattices of varieties

100%
Changphas T., Denecke K.
Discussiones Mathematicae - General Algebra and Applications
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2003
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tom 23
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nr 1
31-43
EN
Hypersubstitutions are mappings which map operation symbols to terms. The set of all hypersubstitutions of a given type forms a monoid with respect to the composition of operations. Together with a second binary operation, to be written as addition, the set of all hypersubstitutions of a given type forms a left-seminearring. Monoids and left-seminearrings of hypersubstitutions can be used to describe complete sublattices of the lattice of all varieties of algebras of a given type. The complexity of a hypersubstitution can be measured by the complexity of the resulting terms. We prove that the set of all hypersubstitutions with a complexity greater than a given natural number forms a sub-left-seminearring of the left-seminearring of all hypersubstitutions of the considered type. Next we look to a special complexity measure, the operation symbol count op(t) of a term t and determine the greatest M-solid variety of semigroups where $M = H₂^{op}$ is the left-seminearring of all hypersubstitutions for which the number of operation symbols occurring in the resulting term is greater than or equal to 2. For every n ≥ 1 and for $M = Hₙ^{op}$ we determine the complete lattices of all M-solid varieties of semigroups.
11
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On sets related to maximal clones

100%
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
101-114
EN
For an arbitrary h-ary relation ρ we are interested to express n-clone Polⁿρ in terms of some subsets of the set of all n-ary operations Oⁿ(A) on a finite set A, which are in general not clones but we can obtain Polⁿρ from these sets by using intersection and union. Therefore we specify the concept a function preserves a relation and moreover, we study the properties of this new concept and the connection between these sets and Polⁿρ. Particularly we study $R_{a̲,b}^{n,k}$ for arbitrary partial order relations, equivalence relations and central relations.
12
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The Galois correspondence between subvariety lattices and monoids of hpersubstitutions

81%
Denecke K., Hyndman J., Wismath S.
Discussiones Mathematicae - General Algebra and Applications
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2000
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tom 20
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nr 1
21-36
EN
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.
13
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The semantical hyperunification problem

81%
Denecke K., Koppitz J., Wismath S.
Discussiones Mathematicae - General Algebra and Applications
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2001
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tom 21
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nr 2
175-200
EN
A hypersubstitution of a fixed type τ maps n-ary operation symbols of the type to n-ary terms of the type. Such a mapping induces a unique mapping defined on the set of all terms of type t. The kernel of this induced mapping is called the kernel of the hypersubstitution, and it is a fully invariant congruence relation on the (absolutely free) term algebra $F_{τ}(X)$ of the considered type ([2]). If V is a variety of type τ, we consider the composition of the natural homomorphism with the mapping induced by a hypersubstitution. The kernel of this mapping is called the semantical kernel of the hypersubstitution with respect to the given variety. If the pair (s,t) of terms belongs to the semantical kernel of a hypersubstitution, then this hypersubstitution equalizes s and t with respect to the variety. Generalizing the concept of a unifier, we define a semantical hyperunifier for a pair of terms with respect to a variety. The problem of finding a semantical hyperunifier with respect to a given variety for any two terms is then called the semantical hyperunification problem. We prove that the semantical kernel of a hypersubstitution is a fully invariant congruence relation on the absolutely free algebra of the given type. Using this kernel, we define three relations between sets of hypersubstitutions and sets of varieties and introduce the Galois correspondences induced by these relations. Then we apply these general concepts to varieties of semigroups.
14
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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.
15
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The greatest regular-solid variety of semigroups % Dedicated to R. McKenzie's 60$^th$ birthday %

81%
Denecke K., Koppitz J., Pabhapote N.
Discussiones Mathematicae - General Algebra and Applications
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2008
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tom 28
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nr 1
91-119
EN
A regular hypersubstitution is a mapping which takes every $n_i$-ary operation symbol to an $n_i$-ary term. A variety is called regular-solid if it contains all algebras derived by regular hypersubstitutions. We determine the greatest regular-solid variety of semigroups. This result will be used to give a new proof for the equational description of the greatest solid variety of semigroups. We show that every variety of semigroups which is finitely based by hyperidentities is also finitely based by identities.
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