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There are three types of Smarandache Algebraic Structures:

1.      A Smarandache Strong Structure on a set S means a structure on S that has a proper subset P with a stronger structure.

1. A Smarandache Weak Structure on a set S means a structure on S that has a proper subset P with a weaker structure.
2. A Smarandache Strong-Weak Structure on a set S means a structure on S that has two proper subsets: P with a stronger structure, and Q with a weaker structure.

By proper subset of a set S, one understands a subset P of S, different from the empty set, from the original set S, and from the idempotent elements if any.

Having two structures {u} and {v} defined by the same operations, one says that structure {u} is stronger than structure {v}, i.e. {u} > {v}, if the operations of {u} satisfy more axioms than the operations of {v}.

Each one of the first two structure types is then generalized from a 2-level (the sets P ⊂ S and their corresponding strong structure {w1}>{w0}, respectively their weak structure {w1}<{w0}) to an n-level (the sets Pn-1 ⊂ Pn-2 ⊂ … ⊂ P2 ⊂ P1 ⊂ S and their corresponding strong structure {wn-1} > {wn-2} > … > {w2} > {w1} > {w0}, or respectively their weak structure {wn-1} < {wn-2} < … < {w2} < {w1} < {w0}). Similarly for the third structure type, whose generalization is a combination of the previous two structures at the n-level.

            A Smarandache Weak BE-Algebra X is a BE-algebra in which there exists a proper subset Q such that 1img src="file:///C:\Users\FLOREN~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.png" height="13" width="13"Q, |Q| ≥ 2, and Q is a CI-algebra.

            And a Smarandache Strong CI-Algebra X is a CI-algebra X in which there exists a proper subset Q such that 1img src="file:///C:\Users\FLOREN~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.png" height="13" width="13"Q, |Q| ≥ 2, and Q is a BE-algebra.

The book elaborates a recollection of the BE/CI-algebras, then introduces these last two particular structures and studies their properties.