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1

Weakly idempotent lattices and bilattices, non-idempotent Plonka functions

Davidova D. S., Movsisyan Y. M.

Demonstratio Mathematica

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2015

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Vol. 48, nr 4

509--535

EN

In this paper, we study weakly idempotent lattices with an additional interlaced operation. We characterize interlacity of a weakly idempotent semilattice operation, using the concept of hyperidentity and prove that a weakly idempotent bilattice with an interlaced operation is epimorphic to the superproduct with negation of two equal lattices. In the last part of the paper, we introduce the concepts of a non-idempotent Plonka function and the weakly Plonka sum and extend the main result for algebras with the well known Plonka function to the algebras with the non-idempotent Plonka function. As a consequence, we characterize the hyperidentities of the variety of weakly idempotent lattices, using non-idempotent Plonka functions, weakly Plonka sums and characterization of cardinality of the sets of operations of subdirectly irreducible algebras with hyperidentities of the variety of weakly idempotent lattices. Applications of weakly idempotent bilattices in multi-valued logic is to appear.

2

Left symmetric left distributive magmas and hypersubstitutions

Vanzurova A.

Demonstratio Mathematica

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2006

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Vol. 39, nr 3

491-506

EN

Our aim is to generalize results reached in [A-D 98] and [V 04]. In [A-D 98], normal forms for terms with respect to the variety SIE of right symmetric idempotent entropic magmas are used to derive multiplication in the magma of normal form hypersubstitutions with respect to SIE, the monoid of SIE-proper normal form hypersubstitutions is found, and hyperidentities are discussed. In [V 04], a similar project is solved for the variety SID of left symmetric left distributive idempotent magmas (in which the variety dual to SIE is contained as a subvariety). Droppping idempotency we obtain a generalization, the variety SD of left symmetric left distributive magmas. We use again (naturally arising) normal forms for terms in SD to study the magma of SD-normal form hypersubstitutions, its multiplication (with six idempotents), and describe the monoid of SD-proper normal form hypersubstitutions. Comparison with the previous cases might be interesting.

3

The pre-clone of a variety

Denecke K., Wismath S.

Demonstratio Mathematica

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2005

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Vol. 38, nr 4

799--810

EN

Clones are sets of operations on a given base set which are closed under superposition of operations and which contain all the projection operations. A clone can be regarded as a heterogeneous or multi-based algebra, consisting of universe sets of n-ary operations, for n > 1, with the superposition operations Snm, for n,m > 1. Such an algebra is called a Menger system. For a fixed n, a set of n-ary operations with the (n + 1)-ary superposition operation Sn forms a homogeneous algebra called a Menger algebra of rank n. These structures can also be defined on sets of term operations of a fixed type r. The set of all terms of type r which contain at least one operation symbol forms such a clone-like structure which we call the pre-clone of type r. We determine a generating system for this pre-clone. A similar structure can be formed using the terms with respect to a variety, giving the pre-clone of a variety. The normalization of a variety is the model class of all identities s~~t of the variety for which both s and t contain at least one operation symbol. We show that identities in the pre-clone of the normalization of a variety correspond to normal hyperidentities in the normalization.