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On the structure of semilattice sums. (English) Zbl 0793.08010
This paper investigates the structure of algebras of given type $$\tau: \Omega\to N$$ in regular classes and in particular in regular classes of modes [see the authors: Modal theory. An algebraic approach to order, geometry, and convexity (1985; Zbl 0553.08001)]. Recall that an identity is regular if the sets of variables appearing on each side are equal. A class $${\mathcal V}$$ of algebras of type $$\tau: \Omega\to N$$ is regular if the only identities satisfied by all algebras in $${\mathcal V}$$ are regular. Otherwise, $${\mathcal V}$$ is irregular. The algebras considered in this paper are of type $$\tau: \Omega\to \mathbb{Z}^ +$$ with $$2\in \tau(\Omega)$$.

##### MSC:
 08C99 Other classes of algebras 06A12 Semilattices 08B99 Varieties 08B25 Products, amalgamated products, and other kinds of limits and colimits
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##### References:
 [1] A. H. Clifford: Semigroups admitting relative inverses. Ann. Math. 42 (1941), 1037-1049. · Zbl 0063.00920 [2] P. M. Cohn: Universal Algebra. Reidel, Dordrecht, 1981. · Zbl 0461.08001 [3] J. Dudek: Varieties of idempotent commutative groupoids. Fund. Math. 120 (1983), 193-204. · Zbl 0546.20049 [4] J. Dudek E. Graczyńska: The lattice of varieties of algebras. Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), 337-340. [5] E. Graczyńska D. Kelly P. Winkler: On the regular part of varieties of algebras. Algebra Universalis 23 (1986), 77-84. · Zbl 0605.08004 [6] J. Ježek T. Kepka: Ideal free CIM-groupoids and open convex sets. Springer Lecture Notes in Mathematics 1004, Universal Algebra and Lattice Theory, 1983, 166-175. [7] J. Ježek T. Kepka: Medial Groupoids. Rozpravy ČSAV, Řada mat. a přír. věd. 93/2 (1983), Academia, Praha. · Zbl 0538.08008 [8] J. Ježek T. Kepka: The lattice of varieties of commutative idempotent abelian distributive groupoids. Algebra Universalis 5 (1975), 225-237. · Zbl 0321.20036 [9] M. Kuczma: An introduction to the Theory of Functional Equations and Inequalities. P.W.N, Warszawa, 1985. · Zbl 0555.39004 [10] G. Lallement: Demi-groupes réguliers. Ann. Math. Pura Appl. 77 (1967), 47-129. · Zbl 0203.02001 [11] A. I. Maľcev: Algebraic Systems. Springer-Verlag, Berlin 1973. · Zbl 0282.65028 [12] A. I. Maľcev: Multiplication of classes of algebraic systems. (Russian), Sibirsk. Math. Zh. 8 (1967), 346-365. [13] I. I. Meľnik: Normal closures of perfect varieties of universal algebras. (Russian), Ordered sets and lattices, Izdat. Saratov. Univ., Saratov (1971), 56-65, MR 52 # 3011. [14] B. Melnik: Theory of filters. Commun. Math. Phys. 15 (1969), 1 - 46. · Zbl 0182.59601 [15] W. D. Neumann: On the quasivariety of convex subsets of affine spaces. Arch. Math. 21 (1970), 11-16. · Zbl 0194.01502 [16] M. Petrich: Lectures in Semigroups. Akademie-Verlag, Berlin, 1977. · Zbl 0369.20036 [17] J. Plonka: On a method of construction of abstract algebras. Fund. Math. 61 (1967), 183-189. · Zbl 0168.26701 [18] J. Plonka: On equational classes of abstract algebras defined by regular equations. Fund. Math. 64 (1969), 241-247. · Zbl 0187.28702 [19] F. Poyatos: Generalizacion de un teorema de J. A. Green a algebras universales. Rev. Math. Hisp.-Amer. (4)40 (1980), 193-205. · Zbl 0523.08001 [20] A. Romanowska: Constructing and reconstructing of algebras. Demonstratio Math. 18 (1985), 209-230. · Zbl 0594.08001 [21] A. Romanowska: On regular and regularised varieties. Algebra Universalis 23 (1986), 215-241. · Zbl 0619.08004 [22] A. Romanowska J. D. H. Smith: From affine to projective geometry via convexity. Springer Lecture Notes in Mathematics 1149, Universal Algebra and Lattice Theory, 1985, 255-269. [23] A. Romanowska J. D. H. Smith: Modal Theory-an Algebraic Approach to Order. Geometry and Convexity, Heldermann Verlag, Berlin, 1985. · Zbl 0553.08001 [24] A. Romanowska J. D. H. Smith: On the structure of barycentric algebras. Houston J. Math.,toappear. · Zbl 0725.08001 [25] W. W. Saliĭ: Equationally normal varieties of semigroups. (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 84 (1969), 61-68, MR 41 # 8555. [26] W. W. Saliĭ: A theorem on homomorphisms of strong semilattices of semigroups. (Russian), Theory of semigroups and its applications, V. V. Vagner (ed.), Izd. Saratov. Univ. 2 (1970), 69-74, MR 53 # 10959.
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