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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 Γ(L)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma (L)$$\end{document}. This fact plays an important role. Namely, PCS L is a disjoint union of all congruence classes of Γ(L)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma (L)$$\end{document} (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 {Γa∣a∈B(L)}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{\,\Gamma _a\mid a\in B(L)\,\}$$\end{document} and a suitable semilattice homomorphism φa,b:Γa→Γb\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi _{a,b}:\Gamma _a\rightarrow \Gamma _b$$\end{document} for any a≥b\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a\ge b$$\end{document} 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∈D(L)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1\in D(L)$$\end{document}. 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.
algebra universalis – Springer Journals
Published: Nov 1, 2021
Keywords: Pseudocomplemented semilattice; Closed elements; Dense element; Glivenko-Frink congruence; GF-block; Associated full triple; Abstract full triple; 06A12; 06D15; 06D20; 08A05; 08A30
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