Near unanimity: an obstacle to general duality theory

Until now, papers on duality theory within general algebra have concentrated on finding conditions on a finite non-trivial algebra P which will allow us to prove easily that there is a natural duality for the quasivariety d .'= nNP(P) generated by P (see [9; 10; 11;4]) or, more recently, on ways to refine an existing duality by deleting unnecessary structure from the objects in the dual category (see C5; 6]). This paper begins the attack on a much more fundamental question: " W h i c h f i n i t e a lgebras P a d m i t a n a t u r a l d u a l i t y ? " (We say that _P a d m i t s a na tu ra l dua l i t y or, when we are being more colloquial, is dual izable , if there is a natural duality for the quasivariety generated by _P.) Davey and Werner [9] showed that if _P has a n e a r u n a n i m i t y term, that is, a term t of arity at least three such that _P satisfies the identities