# Article

Full entry | PDF   (0.8 MB)
Keywords:
distributive double \$p\$-algebra; variety; endomorphism monoid; equimorphy; categorical universality
Summary:
Any finitely generated regular variety \$\mathbb{V}\$ of distributive double \$p\$-algebras is finitely determined, meaning that for some finite cardinal \$n(\mathbb{V})\$, any subclass \$S\subseteq \mathbb{V}\$ of algebras with isomorphic endomorphism monoids has fewer than \$n(\mathbb{V})\$ pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double \$p\$-algebras must be almost regular.
References:
[1] M. E. Adams, V. Koubek and J. Sichler: Homomorphisms and endomorphisms in varieties of pseudocomplemented distributive lattices (with applications to Heyting algebras). Trans. Amer. Math. Soc. 285 (1984), 57–79. DOI 10.1090/S0002-9947-1984-0748830-6 | MR 0748830
[2] M. E. Adams, V. Koubek and J. Sichler: Pseudocomplemented distributive lattices with small endomorphism monoids. Bull. Austral. Math. Soc. 28 (1983), 305–318. DOI 10.1017/S0004972700021031 | MR 0729763
[3] R. Beazer: The determination congruence on double \$p\$-algebras. Algebra Universalis 6 (1976), 121–129. DOI 10.1007/BF02485824 | MR 0419319 | Zbl 0353.06002
[4] B. A. Davey: Subdirectly irreducible distributive double \$p\$-algebras. Algebra Universalis 8 (1978), 73–88. DOI 10.1007/BF02485372 | MR 0450160 | Zbl 0381.06019
[5] L. M. Gluskin: Semigroups of isotone transformations. Uspekhi Math. Nauk 16 (1961), 157–162. (Russian) MR 0131486
[6] V. Koubek: Infinite image homomorphisms of distributive bounded lattices. Coll. Math. Soc. János Bolyai, 43. Lecture in Universal Algebra, Szeged 1983, North Holland, Amsterdam, 1985, pp. 241–281. MR 0860268
[7] V. Koubek and H. Radovanská: Algebras determined by their endomorphism monoids. Cahiers Topologie Gèom. Différentielle Catégoriques 35 (1994), 187–225. MR 1295117
[8] V. Koubek and J. Sichler: Universal varieties of distributive double \$p\$-algebras. Glasgow Math. J. 26 (1985), 121–131. DOI 10.1017/S0017089500005887 | MR 0798738
[9] V. Koubek and J. Sichler: Categorical universality of regular distributive double \$p\$-algebras. Glasgow Math. J. 32 (1990), 329–340. DOI 10.1017/S0017089500009411 | MR 1073673
[10] V. Koubek and J. Sichler: Finitely generated universal varieties of distributive double \$p\$-algebras. Cahiers Topologie Gèom. Différentielle Catégoriques 35 (1994), 139–164. MR 1280987
[11] V. Koubek and J. Sichler: Priestley duals of products. Cahiers Topologie Gèom. Différentielle Catégoriques 32 (1991), 243–256. MR 1158110
[12] K. D. Magill: The semigroup of endomorphisms of a Boolean ring. Semigroup Forum 4 (1972), 411–416. MR 0272690
[13] C. J. Maxson: On semigroups of Boolean ring endomorphisms. Semigroup Forum 4 (1972), 78–82. DOI 10.1007/BF02570772 | MR 0297900 | Zbl 0262.06011
[14] R. McKenzie and C. Tsinakis: On recovering a bounded distributive lattices from its endomorphism monoid. Houston J. Math. 7 (1981), 525–529. MR 0658568
[15] H. A. Priestley: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2 (1970), 186–190. DOI 10.1112/blms/2.2.186 | MR 0265242 | Zbl 0201.01802
[16] H. A. Priestley: The construction of spaces dual to pseudocomplemented distributive lattices. Quart. J. Math. Oxford 26 (1975), 215–228. DOI 10.1093/qmath/26.1.215 | MR 0392731 | Zbl 0323.06013
[17] H. A. Priestley: Ordered sets and duality for distributive lattices. Ann. Discrete Math. 23 (1984), 36–60. MR 0779844 | Zbl 0557.06007
[18] A. Pultr and V. Trnková: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North Holland, Amsterdam, 1980. MR 0563525
[19] B. M. Schein: Ordered sets, semilattices, distributive lattices and Boolean algebras with homomorphic endomorphism semigroups. Fund. Math. 68 (1970), 31–50. DOI 10.4064/fm-68-1-31-50 | MR 0272686 | Zbl 0197.28902

Partner of