Die messianische Weissagung und ihre Erfüllung

Structure of relatively free n-tuple semigroups

“…More precisely, a cubical dialgebra [15] is a vector space A equipped with two binary operations: ⊣ called left and ⊢ called right, satisfying (3.1) for any • 1 , • 2 ∈ {⊣, ⊢}. This structure under the name "doppelalgebra" has been considered earlier by Richter [17].…”

“…The study of doppelsemigroups was initiated by the author in [28] and then it was continued in [5,6,25,30,33,35,36,41,44]. Note that doppelalgebras introduced by Richter [17] are linear analogs of doppelsemigroups. These objects have been characterized in [1] as algebras over some operads.…”

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]…”

“…A doppelsemigroup (see [6,129,225,243,308,310]) is an algebra with two binary operations satisfying the following hyperidentity of associativity X(x, Y (y, z)) = Y (X(x, y), z).…”