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DEMONSTRATIO MATHEMATICAVol. XLINo 22008Jerzy Ρ tonkaSUBDIRECT PRODUCT REPRESENTATIONSOF SOME UNARY EXTENSIONS OF SEMILATTICESAbstract. An algebra 21 represents the sequence so = (0,3,1,1,...) if there are noconstants in 21, there are exactly 3 distinct essentially unary polynomials in 2t and exactly1 essentially n-ary polynomial in 21 for every η > 1. It was proved in [4] that an algebra 21represents the sequence so if and only if it is clone equivalent to a generic of one of threevarieties Vi, V2, V3, see Section 1 of [4]. Moreover, some representations of algebras fromthese varieties by means of semilattice ordered systems of algebras were given in [4], Inthis paper we give another, by subdirect products, representation of algebras from Vi, V2,V3. Moreover, we describe all subdirectly irreducible algebras from these varieties and weshow that if an algebra 21 represents the sequence so, then it must be of cardinality atleast 4.1. PreliminariesBy an algebra we mean a pair 21 = (A; Fα), where A is a nonemptyset called the carrier of 21 and F21 is a set of finitary operations in 21called the set of fundamental operations of 21. By the clone of 21 we meanas usually the smallest set containing all projections
Demonstratio Mathematica – de Gruyter
Published: Apr 1, 2008
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