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On a new construction of pseudocomplemented semilattices

Abstract

In the theory of semigroups there exists a construction of some semilattices from a special family of semigroups. It is the so-called ‘strong semilattice of semigroups’. Modifying this idea we can present a new construction method for arbitrary pseudocomplemented semilattice (= PCS) L using ‘full triples’. This construction centers around the classical Glivenko-Frink congruence \(\Gamma (L)\). This fact plays an important role. Namely, PCS L is a disjoint union of all congruence classes of \(\Gamma (L)\) (or of GF-blocks for short). In order to get a ‘full triple’ of L, the so-called ‘associate’ full triple of L, we need the Boolean algebra of closed elements B(L), the whole family of GF-blocks \(\{\,\Gamma _a\mid a\in B(L)\,\}\) and a suitable semilattice homomorphism \(\varphi _{a,b}:\Gamma _a\rightarrow \Gamma _b\) for any \(a\ge b\) in B(L). There is also a definition of an ‘abstract’ full triple, which we use by a construction of a PCS. The notion of a full triple is an extension of the ‘classical’ triple, which do work only with just one GF-block D(L) satisfying \(1\in D(L)\). It is known that there exist PCS’s which cannot be constructed by using a classical triple method. In addition, we explore in some detail the homomorphisms and the subalgebras of PCS’s.

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Acknowledgements

Our thanks go the referee who read the paper with extraordinary care and patience. His/her constructive criticisms have been invaluable.

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Correspondence to Tibor Katriňák.

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Dedicated to the memory of R. Wille, K. Keimel and P. Burmeister.

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While working on this paper, both authors were supported by VEGA grant No. 1/0333/17 of Slovak Republic.

Communicated by Presented by M. Haviar.

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Katriňák, T., Guričan, J. On a new construction of pseudocomplemented semilattices. Algebra Univers. 82, 54 (2021). https://doi.org/10.1007/s00012-021-00746-1

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Keywords

  • Pseudocomplemented semilattice
  • Closed elements
  • Dense element
  • Glivenko-Frink congruence
  • GF-block
  • Associated full triple
  • Abstract full triple

Mathematics Subject Classification

  • 06A12
  • 06D15
  • 06D20
  • 08A05
  • 08A30