Code theory and fuzzy subsemigroups. (English) Zbl 0634.94027

Let L be a lattice and \(B^+\) the free semigroup with set generator B. Then A. Rosenfeld [J. Math. Anal. Appl. 35, 512-517 (1971; Zbl 0194.055)] calls fuzzy subsemigroup any map \(s:B^+\to L\) such that the cut \(C^ u_ s=\{x\in B^+/s(x)\geq u\}\) is a subsemigroup for every element u of L. This is equivalent requiring that \(s(xy)\geq s(x)\wedge s(y)\) for every x, y in L.
In this paper one calls free every fuzzy subsemigroup s of \(B^+\) whose cuts are free subsemigroups. One observe that this is equivalent to require that \(s(x)\geq s(yx)\wedge s(xy)\wedge s(y).\) Analogous definitions and characterizations are given for the pure, very pure, left unitary, right unitary, unitary fuzzy subsemigroups. Moreover, severals methods for constructing fuzzy subsemigroups of such type are given [see also L. Biacino and G. Gerla, Inf. Sci. 32, 181-195 (1984; Zbl 0562.06004)].
Reviewer: G.Gerla


94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
20M05 Free semigroups, generators and relations, word problems
94A15 Information theory (general)
68Q45 Formal languages and automata
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