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On the structure of subalgebra systems of idempotent entropic algebras. (English) Zbl 0663.08002
The paper continues the study of subalgebra systems initiated in the preceding one (see Zbl 0663.08001). In the present part the subalgebra systems of idempotent entropic algebras (or IEO-semilattices) are investigated by means of various decomposition methods. The attention is focused on the decomposition of meet-distributive bisemilattices since these algebras play an essential role in the structure of IEO- semilattices. Simultaneously the corresponding methods for recovering the original idempotent entropic algebras from their decompositions are given. It is shown that the generalized Płonka sum is a useful tool for reconstruction of some idempotent entropic algebras.
Reviewer: J.Duda

MSC:
08A30 Subalgebras, congruence relations
06A12 Semilattices
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References:
[1] Aigner, M, Kombinatorik, I, (1975), Springer Berlin/Heidelberg/New York · Zbl 0373.05001
[2] Birkhoff, G, Lattice theory, (1967), Amer. Math. Soc Providence, RI · Zbl 0126.03801
[3] Bruck, R.H, A survey of binary systems, (1958), Springer Berlin/Heidelberg/New York · Zbl 0141.01401
[4] Cohn, P.M, Universal algebra, (1965), Harper and Row New York · Zbl 0141.01002
[5] {\scJ. Dudek}, Varieties of idempotent commutative groupoids, manuscript. · Zbl 0546.20049
[6] Grätzer, G; Plonka, J, A characterization of semilattices, (), 21-24 · Zbl 0216.03401
[7] Howie, J.M, An introduction to semigroup theory, (1976), Academic Press London · Zbl 0355.20056
[8] {\scJ. Ježek and T. Kepka}, Idealfree CIM-groupoids and open convex sets, manuscript. · Zbl 0524.20031
[9] Jónsson, B; Nelson, E, Relatively free products in regular varieties, Algebra universalis, 4, 14-19, (1974) · Zbl 0319.08002
[10] Kimura, N; Yamada, M, Note on idempotent semigroups, II, (), 110-112 · Zbl 0081.25501
[11] Mac Lane, S, Categories for the working Mathematician, (1971), Springer New York · Zbl 0232.18001
[12] Mal’cev, A.I, Algebraic systems, (1973), Springer Berlin/Heidelberg/New York
[13] Mitschke, A, On a representation of groupoids as sums of directed systems, (), 11-18 · Zbl 0277.20074
[14] Plonka, J, Diagonal algebras, Fund. math., 58, 309-321, (1966) · Zbl 0166.27404
[15] Plonka, J, Remarks on diagonal and generalized diagonal algebras, (), 19-23 · Zbl 0166.27501
[16] Plonka, J, On a method of construction of abstract algebras, Fund. math., 61, 183-189, (1967) · Zbl 0168.26701
[17] Plonka, J, A representation theorem for idempotent medial algebras, Fund. math., 61, 191-198, (1967) · Zbl 0164.01102
[18] Plonka, J, On equational classes of abstract algebras defined by regular equations, Fund. math., 64, 241-247, (1969) · Zbl 0187.28702
[19] Romanowska, A, Subdirectly irreducible.-distributive bisemilattices, I, Demonstratio math., 13, 767-785, (1980) · Zbl 0448.08007
[20] Romanowska, A.B; Smith, J.D.H, Bisemilattices of subsemilattices, J. algebra, 70, 78-88, (1981) · Zbl 0457.06002
[21] Romanowska, A.B; Smith, J.D.H, Subalgebra systems of idempotent entropic algebras, J. algebra, 120, 247-262, (1989) · Zbl 0663.08001
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