A characterization of varieties with a difference term

ABSTRACT. We provide more characterizations of varieties with a weak difference term and of neutral varieties. We prove that a variety has a (weak) difference term (is neutral) with respect to the TC-commutator iff it has a (weak) difference term (is neutral) with respect to the linear commutator. We show that a variety V is congruence meet semi-distributive iff V is neutral, iff M3 is not a sublattice of Con A, forA2V, iff there is a positive integer n such that V j ≥ Con ã(å Žç) ãån. 1. Preliminaries. In this section we recall some definitions. The rest of the paper is independent from this section, provided the reader is acquainted with the relevant definitions. Familiarity with Part I [Lp1] is welcome, but probably not indispensable for reading the paper. The results presented here and their connection with part I are discussed at the beginning of each section. Basics about universal algebra can be found, e.g., in [BS], [MKNT]. Let A be an algebra, and ã, å, ç be congruences on A. M(ã, å) denotes the set of all matrices of the form: þ t(ā, ¯b) t(ā, ¯b 0) t(ā0, ¯b) t(ā0, ¯b 0) þ where ā, ā0 2 An, ¯b, ¯b 0 2 Am,forsomen,m0, t is any m + n-ary term operation of A, and āãā0, ¯bå¯b 0. þ 2 M(ã, å), and açb then also cçd. C(ã, å; ç) means that if þ a b c d The (TC) commutator [ã, å]ofãand å is the least congruence ç such that C(ã, å; ç) holds. The symmetric commutator [ã, å]sym of ã and å is the least congruence ç such that both C(ã, å; ç) andC(å,ã;ç) hold. Another commutator will play an important role in this paper, the linear commutator [,]L (Definitions 2.1). Other commutators have some interest [Lp2]; in Part I we used another intermediate commutator [ ,]2T; in the present paper we show that its use can be avoided. We have that [ ,]sym,[,]2Tand [ ,]L are all symmetric operations, that [ã, å