On Algebraic Multi-group Spaces¸linfan

A Smarandache multi-space is a union of n spaces A 1 , A 2 , · · · , A n with some additional conditions holding. Combining classical of a group with Smarandache multi-spaces, the conception of a multi-group space is introduced in this paper, which is a generalization of the classical algebraic structures, such as the group, filed, body, · · ·, etc.. Similar to groups, some characteristics of a multi-group space are obtained in this paper. The notion of multi-spaces is introduced by Smarandache in [5] under his idea of hybrid mathematics: combining different fields into a unifying field([6]). Today, this idea is widely accepted by the world of sciences. For mathematics, definite or exact solution under a given condition is not the only object for mathematician. New creation power has emerged and new era for the mathematics has come now. A Smarandache multi-space is defined by Definition 1.1 For any integer i, 1 ≤ i ≤ n let A i be a set with ensemble of law L i , and the intersection of k sets A i 1 , A i 2 , · · · , A i k of them constrains the law A = n i=1 A i is called a multi-space. The conception of multi-group space is a generalization of the classical algebraic structures, such as the group, filed, body, · · ·, etc., which is defined as follows. Definition 1.2 Let G = n i=1 G i be a complete multi-space with a binary operation set O(G) = {× i , 1 ≤ i ≤ n}. If for any integer i, 1 ≤ i ≤ n, (G i ; × i) is a group and for ∀x, y, z ∈ G and any two binary operations × and • , × = • , there is one operation,