A nonassociative extension of the class of distributive lattices

Ervin Fried and George Grätzer

Vol. 49 (1973), No. 1, 59–78

DOI: 10.2140/pjm.1973.49.59

Abstract

Let Z = {0,1,2} and define two binary operations ∧ and V on Z as follows: 0Λ1 = 0,0 vl = 1,1 ∧ 2 = 1, lv2 = 2,2 ∧ 0 = 2,2 ∨ 0 = 2, both operations are idempotent and commutative. This paper deals with the equational class Z generated by the algebra ⟨Z;∧,∨⟩. The class Z contains the class of all distributive lattices and Z is a subclass of the class of weakly associative lattices (trellis, T-lattice) in the sense of E. Fried and H. Skala.

The purpose of this paper is to prove that Z shares the most important properties of the class of distributive lattices.

Mathematical Subject Classification

Primary: 06A20

Milestones

Received: 19 July 1972

Revised: 15 October 1972

Published: 1 November 1973