Implication Zroupoids and Identities of Associative Type

JUAN MANUEL CORNEJO; HANAMANTAGOUDA P. SANKAPPANAVAR

Quasigroups and Related Systems

Institute of Mathematics of the Moldovian Academy of Sciences

Año: 2018 vol. 26 p. 13 - 34

1561-2848

An algebra A = hA;!; 0i, where ! is binary and 0 is a constant,is called an implication zroupoid (I-zroupoid, for short) if A satisesthe identities: (x ! y) ! z [(z0 ! x) ! (y ! z)0]0 and 000 0,where x0 := x ! 0, and I denotes the variety of all I-zroupoids. An I-zroupoid is symmetric if it satises x00 x and (x ! y0)0 (y ! x0)0.The variety of symmetric I-zroupoids is denoted by S. An identityp q, in the groupoid language h!i, is called an identity of associativetype of length 3 if p and q have exactly 3 (distinct) variables, say x,y,z,and are grouped according to one of the two ways of grouping: (1)? ! (? ! ?) and (2) (? ! ?) ! ?:, where ? is a place holder for avariable.In this paper we give a complete analysis of the mutual relation-ships of all subvarieties of I, each of which is dened by a singleidentity of associative type of length 3. We prove, in our main theo-rem, that there are exactly 8 such subvarieties of I that are distinctfrom each other and describe explicitly the poset formed by them un-der inclusion. As an application of the main theorem, we derive thatthere are three distinct subvarieties of the variety S, each dened bya single identity of associative type of length 3.