Semigroups with n + 1 Distinct Essentially n-Ary Term Operations

Square extension of a groupoid

“…In [16] and [18] semigroups with exactly n + 1 different essentially n-ary term operations for every positive integer n were considered. The following characterization theorem was proved.…”

“…Theorem 4.6 (cf. [18], Theorem, Section 2). If a semigroup represents the sequence (0, 2, 3, 4), then it is noncommutative and belongs to one of two up to duality uniquely determined varieties K 1 , K 2 defined respectively by the following sets of identities:…”

“…Observe that, according to the result of [4], the varieties defined above are log-linear since, as the reader can check, the number of different n-ary term operations over a groupoid having exactly n + 1 different essentially n-ary term operations is equal to (n + 2)2 n−1 − 1 for every non-negative integer n. Note also that the groupoids investigated in [18] have polynomially bounded p n -sequences but they do not need to be finite. Now, our aim is to give a structural description of the groupoids from the variety K 1 .…”

“…The method of construction presented in this paper gives an easy way of studying groupoid structures, since for a given variety of groupoids, by a simple modification of identities considered in our paper and using the method presented here, one can obtain required representation theorem. This paper originates from the investigation of a structural characterization of algebras considered by the first author, see [18]. This determines the identities considered here.…”