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2 Płonka J.

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Subdirect product representations of some unary extensions of semilattices

Płonka J.

Demonstratio Mathematica

2008

Vol. 41, nr 2

263-272

An algebra [...] represents the sequence so = (0, 3, l, l, . . .) if there are no constants in [...], there are exactly 3 distinct essentially unary polynomials in [...] and exactly l essentially n-ary polynomial in [...] for every n > l . It was proved in [4] that an algebra [..] represents the sequence so if and only if it is clone equivalent to a generic of one of three varieties V1, V2, V3, see Section l of [4]. Moreover, some representations of algebras from these varieties by means of semilattice ordered systems of algebras were given in [4] . In this paper we give another, by subdirect products, representation of algebras from V1, V2, V3. Moreover, we describe all subdirectly irreducible algebras from these varieties and we show that if an algebra [...] represents the sequence so, then it must be of cardinality at least 4.

Left-outtermost extension of some varieties

Płonka J.

Demonstratio Mathematica

2003

Vol. 36, nr 1

37--51

Let r : F - > N be a type of algebras F is a nonempty set of fundamental operation symbols and N is the set of all positive integers. An identity ip fa if) of type T we call left-outermost if the left-outermost variables in ip and ip are the same. For a variety V of type r we denote by Vi the variety of type r defined by all left-outermost identities from Id(V). Vl is called the left-outermost extension of V. In this paper we study minimal generics, subdirectly irreducible algebras and lattices of subvarieties in left-outermost extensions of some generalizations of the variety D of all distributive lattices.