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Zbl 0722.06004
Mittas, Jean; Konstantinidou, Maria
Sur une nouvelle génération de la notion de treillis. Les supertreillis et certaines de leurs propriétés générales. (On a new generalization of the notion of a lattice. Superlattices and some of their general properties). (French)
[J] Ann. Sci. Univ. Blaise Pascal Clermont-Ferrand II 94, Math. 25, 61-83 (1989). ISSN 0249-7042

The authors consider a nonempty set S endowed with two multioperations $a\vee b$ and $a\wedge b$ (the value of $a\vee b$ is a subset of S, and similarly for $a\wedge b)$ satisfying certain axioms; the structure obtained in this way is called a superlattice. Particular cases of superlattices are (i) lattices; (ii) multilattices [cf. {\it M. Benado}, Czech. Math. J. 5(80), 308-344 (1955; Zbl 0068.259)]; (iii) hyperlattices [cf. the authors, Math. Balk. 7, 187-193 (1977; Zbl 0486.06004)]. In the first part of the paper the basic properties of superlattices are deduced; typical result: a superlattice (S,$\wedge,\vee)$ is a lattice if and only if $(a\vee b)\wedge a=(a\wedge b)\vee a=a$ for each a,b$\in S$. Several examples and some additional results are given in the second part of the article. \par [In the title of the paper it should evidently be ``généralisation'' instead of ``génération''.]
[J.Jakubík (Košice)]

MSC 2000:: *06B99 Lattices

Keywords: multioperations; multilattices; hyperlattices; superlattices

Citations: Zbl 0068.259; Zbl 0486.06004

Cited in: Zbl 0804.06011

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