eO-代数簇公理.pdfVIP

Journal of Mathematical Research Exposition Aug., 2008, Vol. 28, No. 3 pp. 699–705 DOI:10.3770/j.issn:1000-341X.2008.03.034 Http:// Axioms in the Variety of eO-Algebras FANG Jie1,2, SUN Zhong Ju1 (1. Department of Mathematics, Shantou University, Guangdong 515063, China; 2. Faculty of Mathematics and Computer Science, Guangdong Polytechnic Normal University, Guangdong, 510665, China) (E-mail: jfang@; g zjsun@) Abstract The variety eO of extended Ockham algebras consists of those algebras (L; ∧, ∨, f, k, 0, 1) such that (L; ∧, ∨, 0, 1) is a bounded distributive lattice together with a dual endomor- phism f on L and an endomorphism k on L such that fk = kf . In this paper we extend Urquhart’s theorem to eO-algebras and we are in particular concerned with the subclass e2M of eO-algebras in which f2 = id and k2 = id. We show that there are 19 non-equivalent axioms in e2M and then order them by implication. Keywords Extended Ockham algebra; dual space; subdirectly irreducible algebra; equational basis. Document code A MR(2000) Subject Classification 06D15; 06D30 Chinese Library Classification O153.1 1. Introduction In [3], we introduced the notion of extended Ockham algebra. By an extended Ockham algebra (L; ∧, ∨, f, k, 0, 1), we mean a bounded distributive lattice L on which two operations f and k are defined such that (1) f is a dual lattice endomorphism with f (1) = 0 and f (0) = 1; (2) k is a lattice endomorphism with k(1) = 1 and k(0) = 0; (3) f and k commute. The class of such algebras is denoted by eO. In the class of eO-algebras, a particular inter- esting subclass is so called class of symmetric extended de Morgan algebras for which f 2