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|Вычислительная математика. Численный анализ (74)|
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CORNEJO, Juan M.; SANKAPPANAVA, Hanamantagouda P.. Implication zroupoids and identities of associative type. In: Quasigroups and Related Systems. 2018, nr. 1(26), pp. 13-34. ISSN 1561-2848.
|Quasigroups and Related Systems|
|Numărul 1(26) / 2018 / ISSN 1561-2848|
An algebra A = hA;!; 0i, where ! is binary and 0 is a constant, is called an Izroupoid if A satis_es the 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 satis_es x00 _ x and (x ! y0)0 _ (y ! x0)0. The variety of symmetric I-zroupoids is denoted by S. An identity p _ q, in the groupoid language h!i, is called an identity of associative type 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 a variable. A subvariety of I is said to be of associative type of length 3, if it is de_ned, relative to I, by a single identity of associative type of length 3. In this paper we give a complete analysis of the mutual relationships of all subvarieties of I of associative type of length 3. We prove, in our main theorem, that there are exactly 8 such subvarieties of I that are distinct from each other and describe explicitly the poset formed by them under inclusion. As an application of the main theorem, we derive that there are three distinct subvarieties of the variety S of associative type, each de_ned relative to S, by a single identity of associative type of length 3.