الفهرس 
Chapter I: PreliminariesOnly 14 pages are availabe for public view
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Abstract
Two classical problems of group theory (module theory) are: When does a
group G (module M ) decomposes into a direct product (direct sum) of
indecomposable groups (modules)? and; When are two such decompositions unique
up to an isomorphism or up to order of factors? Such decomposition, called a Remak
decomposition is known to exist for groups satisfying the minimal condition on the
direct factors (Robinson [23], 3.3.2(Remak)). The uniqueness problem of a Remak
decomposition of groups has been solved by the classical Krull-Remak-Schmidt
theorems (see Robinson [23], 3.3.8, 3.3.9, 3.3.10).
These kinds of existence and uniqueness theorems hold in other classes of
algebras, for example in modules with finite length and, in general, in every abelian
category (see Faith [18], Theorem 17.16 and Theorem 21.6).
In semigroups, Yamamura [24] has developed a uniqueness of spined product
decomposition of so called orthocrypto groups, which is a certain class of orthodox
semigroups. One of the main perposes in the present work is to show that the class of
Ω-Clifford Semigroups (equivalently partial groups as first appeared in [1], see also
[4]), which is certainly a non-abelian category, behaves very well as generalized
groups when dealing with such kind of problems. This work is based principally on
the investigation developed by Abdallah and El-Lithy [9] on the notion of a direct
product and the existence of a Remak decomposition of Ω-partial groups (Clifford
semigroups) with the minimal (equivalently the maximal) condition on Ω-direct
factors.