Abstract
In this paper we consider the notions of hyperidentities and hypervarieties of a given type τ, without nullary operations, originated by J. Aczel [1], V. D. Belousov [2], W. D. Neumann [12] and W. Taylor [21]. Solid varieties are defined. Their equations correspond to hyperidentities and they form a complete sublattice of the lattice L(τ) of all varieties of type τ. A completeness theorem for hyperidentities is formulated in analogue to G. Birkhoff [3]. The technique of weak isomorphisms, introduced by E. Marczewski and A. Goetz [7] assists us in partially answering a problem of W. Taylor (Problem 4, [21]).
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References
Aczél, J.,Proof of a theorem of distributive type hyperidentities. Algebra Universalis1 (1971) 1–6.
Belousov, V. D.,Systems of quasigroups with generalized identities. Uspehi Mat. Nauk20, (1965), 75–146. English Translation: Russian Math. Surveys20, 75–143.
Birkhoff, G., On thestructure of abstract algebras. Proc. Cambridge Philos. Soc.31 (1935), 433–454.
Bergman, G. M.,Hyperidentities of groups and semigroups. Aequationes Math.23 (1981), 50–65.
Edmonds, C. C.,Varieties generated by semigroups of order four. Semigroup Forum21 (1980), 67–81.
Głazek, K. Michalski, J.,Weak homomorphisms of general algebras, Annales Societatis Math. Pol. Series I: Commentationes Math.XIX (1977), 211–228.
Goetz, A.,On weak isomorphisms and weak homomorphisms of abstract algebras Coll. Math.14 (1966), 163–167.
Grätzer, G.,Universal Algebra, Second Edition. Springer-Verlag, New York 1979.
Graczynska, E.,On regular identities. Algebra Universalis17 (1983), 369–375.
Graczynska, E.,The word problem for regular identities. Beiträge zur Algebra und Geometrie24 (1987), 41–49.
Marczewski, E.,Independence and homomorphisms in abstract algebras. Fundamenta Math.50 (1961), 45–61.
Neumann, W. D.,Mal'cev conditions, spectra and Kronecker product. U. Austral. Math. Soc. (A),25 (1987), 103–117.
Padmanabhan, R. andPenner., P.,Bases of hyperidentities of lattices and semilattices. C.R. Math. Acad. Sci., Canada, Vol. IV. No. 1 (1982), 9–14.
Penner, P.,Hyperidentities of semilattices. Houston J. Math., Vol. 10, No. 1 (1984), 81–108.
Penner, P.,Hyperidentities of lattices and semilattices. Algebra Universalis13 (1981), 307–314.
Płonka, J.,On equational classes of algebras defined by regular identities. Fund. Math.64 (1969), 241–247.
Schweigert, D.,On varieties of clones. Semigroup Forum26 (1983), 275–285.
Schweigert, D.,On weak isomorphisms and equational theories, in:Contribution to General Algebra 3, Proceedings of the Vienna Conference, June 21–24, 1984 — Verlag Holder-Pichler-Gempsky, Wien 1985 — Verlag B.G. Teubner, Stuttgart, pp. 335–340.
Schweigert, D.,On algebras and clones Semigroup Forum5 (1987), 85–99.
Taylor, W.,Equational logic (survey). Houston J. Math.5 (1979), 1–83.
Taylor, W.,Hyperidentities and hypervarieties. Aequationes Math.23 (1981), 30–49.
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This work was done while the first author was visiting the University of Kaiserslautern, under the auspices of the Alexander von Humboldt Foundation.
As a part of this paper was done during the second author's visit to La Trobe University, the financial assistance by ARGS grant B 85154851 and by the DFG is gratefully acknowledged.
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Graczyńska, E., Schweigert, D. Hypervarieties of a given type. Algebra Universalis 27, 305–318 (1990). https://doi.org/10.1007/BF01190711
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DOI: https://doi.org/10.1007/BF01190711