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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

MSC:

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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[1] De Luca, A., On some properties of syntactic semigroup of a very pure subsemigroup, RAIRO inform. theor., 14, No. 1, 39-56, (1980) · Zbl 0437.20053
[2] {\scG. Gerla}, L-subsemigroups of a free semigroup, Rend. Mat., in press. · Zbl 0631.20048
[3] Gerla, G., Pavelka’s fuzzy logic and free L-subsemigroups, Z. math. logik grundlag. math., 31, 123-129, (1985) · Zbl 0584.03015
[4] Gerla, G., Some elementary concepts of L-semigroups theory, Ric. di mat., 33, 53-62, (1984) · Zbl 0582.20037
[5] Coguen, J.A., L-fuzzy sets, J. math. anal. appl., 18, 145-171, (1967)
[6] Lallement, G., Semigroup and combinatorial applications, (1979), Wiley New York · Zbl 0428.20034
[7] Restivo, A., On a question of mcnaughton and papert, Inform. and control, 25, 93-101, (1974) · Zbl 0279.68054
[8] Rosenfeld, A., Fuzzy groups, J. math. anal. appl., 35, 512-517, (1971) · Zbl 0194.05501
[9] Sch├╝tzenberger, M.P., Sur certains sous-demigroups qui interviennent dans un probleme de mathematiques appliques, Publ. sci. univ. alger. ser. A, 6, 85-90, (1959)
[10] Tilson, B., The intersection of free submonoids of a free monoid is free, (), 345-350 · Zbl 0261.20060
[11] Zadeh, L.A., Fuzzy sets, Inform and control, 8, 338-353, (1965) · Zbl 0139.24606
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