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SUBDIRECT PRODUCT REPRESENTATIONS OF SOME UNARY EXTENSIONS OF SEMILATTICES

SUBDIRECT PRODUCT REPRESENTATIONS OF SOME UNARY EXTENSIONS OF SEMILATTICES 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

SUBDIRECT PRODUCT REPRESENTATIONS OF SOME UNARY EXTENSIONS OF SEMILATTICES

Demonstratio Mathematica , Volume 41 (2): 10 – Apr 1, 2008

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Publisher
de Gruyter
Copyright
© by Jerzy Płonka
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2008-0204
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Demonstratio Mathematicade Gruyter

Published: Apr 1, 2008

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