Interassociates of the Free Commutative Semigroup on n Generators

Structure of relatively free n-tuple semigroups

“…Consider a semigroup (D, ⊣) defined on the same set. Recall that (D, ⊣) is an interassociativity of (D, ⊢) [3] if (⊣, ⊢) and (⊢, ⊣) are associative pairs of operations on D. Descriptions of all interassociativities of the free commutative semigroup, the bicyclic semigroup and a monogenic semigroup were presented in [7,8,10]. Some methods of constructing interasociativities of a semigroup were developed in [2].…”

“…, n , satisfying the axioms x r y s z = x r y s z for all x, y, z ∈ G and r, s ∈ n. The term "n-tuple semigroup" appears in the article of Koreshkov [12] as a base of the concept of an n-tuple algebra of associative type, while n > 1 pairwise interassociative semigroups give rise to an n-tuple semigroup. The theory of interassociative semigroups was actively studied (see, e.g., [2,3,7,8,10]). Semigroups and doppelsemigroups are partial cases of n-tuple semigroups, n ∈ N. Conversely, if we have an n-tuple semigroup in which n = 1 (n = 2), then it is a semigroup (doppelsemigroup).…”

Hyperidentities and Related Concepts, II

“…Binary algebras with the hyperidentity of associativity (7) under the name of Γ-semigroups (or gamma-semigroups), doppelsemigroups and doppelalgebras also were considered by various authors [14,130,217,259,261,262,225,243,308,310,6,129,35,36,66,79,80,81,82,83,311] (see earlier papers [250,45]…”

Free left n-dinilpotent dimonoids

“…Obviously, a semigroup (D, ) is an interassociate of a semigroup (D, ) if and only if (D, , ) is a doppelalgebra, and if (D, , ) is a commutative dimonoid, then (D, ) is an interassociate of (D, ). Descriptions of all interassociates of the free commutative semigroup on n generators and of a monogenic semigroup are presented in [3,6], respectively.…”