M-solid quasivarieties
by
Ewa Wanda Graczyńska
Opole University of Technology

Varieties of algebras are classes of algebras defined by sets of identities. A more general concept is the notion of a quasivariety, i.e. a class of algebras defined by a set of implications. Well known Birkhoff's and Mal'cev's type theorems describe such structures by means of some algebraic operators on classes. Among them are direct products, homomorphisms, isomorphisms, substructures, some reduced products etc. Therefore there are two possible ways of treating general algebraic structures: from logical and structural point of view. In our approach we deal with both of them.

Following professor's Dietmar Schweigert ideas from the Technische Universität Kaiserslautern, Germany, we generalized the notion of a quasi-identity to hypequasi-identity and their satisfaction in an algebra or a class of algebras. This new strong satisfaction is called a hypersatisfaction. It is closely related with a new construction of so called derived algebras. This notion is a modification of the definition used by P. Cohn. It makes possible to consider all possible operations in an algebra, which can be defined by the fundamental ones. For example in the structure (N, +) of natural numbers with addition, we can define a new äddition", which duplicates the sum m+n. This trick allows us to consider a wider class of algebras, called by us a derived class of algebras. This gave fundamentals for the notion of a solid variety or a quasivariety of algebras.

It happened that such investigations helped us in solving Problem 4 posed by W. Taylor in his pioneer's paper on hyperidentities and four problems posed in another hyperuniversal bible, namely the monograph of K. Denecke and S.L. Wismath entitled Hyperidentities and clones. The main of them are connected with an intersting interplay of that two approaches: structural and logical.

Many examples show that this strong satisfaction is rather rare in many known structures. Therefore it was generalized it to a weaker notion of so called M-hyperidentity or even more general M-hyperquasi-identity and related notions connected with a choosen monoid M. In some structures it was possible to calculate the particular monoid M, for which the invented notion was the best one. Several applications of that concept were presented already in the monograph of J. Koppitz and K. Denecke M-Solid Varieties of Algebras.

Connections with the lattice of all M-solid sub(quasi)varieties of a given (quasi)variety of algebras showed already a long time ago that a new method of exploring the lattice of all sub(quasi)varieties of a given one was offered, which is in general a very hard problem.

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Date received: May 2, 2007