The following Artin theorem about alternative linear algebras defined on the commutative, associative ring with unity is well-known: in an alternative linear algebra, if $$(a,b,c)=0$$
 , then the subalgebra generated by the elements $$a$$
 , $$b$$
 , and $$c$$
 is associative. In this paper a wide generalization of this classical result is proposed using the concepts of hyperidentity and coidentity. The corresponding universal algebras are referred to as $$g$$
 -algebras.

A Generalization of the Artin Theorem

In this paper we prove Cayley-type theorems for $g$-dimonoids using the left (right) acts of sets and concept of dialgebra. 

Cayley-type theorems for $g$-dimonoids

The concept of $\Gamma $-semigroup is a generalization of semigroup. Let S and $\Gamma $ be two nonempty sets. S is called $\Gamma $-semigroup if there exists a mapping $S\times \Gamma \times S\longrightarrow S$, written as $(a, \alpha , b)\longrightarrow a\alpha b$, satisfying the identity $ (a\alpha b)\beta c$ $=$ $a\alpha (b\beta c) $ for all $a, b, c\in S$ and $ \alpha , \beta \in \Gamma $. This article is a survey of some works published by different authors on $\Gamma $-semigroups.

S. Chattopadhyay

Papers

\(\Gamma \)-Semigroups: A Survey