A Smarandache Multi-space

1) A Smarandache Multi-Space is a non-empty set M such that there exist at least two distinct proper subsets A1 and A2 of M endowed with respectively two distinct spacestructures S1 and S2. By proper subset of M one understands a subset of M that is different from the empty set, from the unit element of any, and from M as well. For example: Let’s consider a set M whose two proper distinct subsets A and B are a Haussdorf space and respectively a Banach space. Then M is a Smarandache multispace. The more subsets A1, A2, …, An of M, with respective distinct space-structures S1, S2, …, Sn, the better approach of real world. A Smarandache Infinite Multi-Space one gets when n is infinite. An interesting case is when A1 A2 … An M, where means “strictly included”, and the corresponding space-structures are S1  S2  … Sn  SM, where  means “strictly stronger”. This is a Smarandache chain.

2) Smarandache Multi-Algebraic Structure: The above definition is adjusted to the abstract algebra structures: I mean the structures S1, S2, …, Sn, SM can be groupoids, semigroups, groups, rings, modules, fields, loops, etc.

Reference: F. Smarandache, “Multi-Space and Multi-Structure”, in “Neutrosophy. Neutrosophic Logic, Set, Probability and Statistics”, American Research Press, 1998.