Algebraic And Combinatorial Results Of Order-preserving Full Contraction Transformation Semigroup

Let be a finite set, the semigroup of full contraction transformation and the subsemigroup of of all order-preserving full contraction transformation semigroup. Several works have been done on algebraic properties of semigroups and results were obtained among these are generating set, the structure of starred Green’s relations in , the local and global U-depth of singular self and strictly partial one-one mappings but the combinatorics properties of has not be considered. Therefore, this study focuses on combinatorics properties of using the classes of the starred Green’s relations and other algebraic properties such as the local U-depth, status of which as not be investigated were examined. The structure of Green’s relations of were also examined, which extended some results in the literature. The aim of this study is to develop the algebraic and combinatorics properties of order-preserving full contraction transformation semigroup and objectives are to: (i) determine the local and global U-depth of , where is the generating set; (ii) obtain the status of using the global U-depth; (iii) examine the number of , , and classes of height r ; (iv) determine the total number of , , and classes of ; (v) investigate the number of elements in each , , and classes within ; and (vi) characterize Green’s relations of . The following procedures were used to obtained the results of the study: the elements of the semigroup were arranged based on their height, within each height by their images sets and their kernel sets; from the table obtained triangular array and sequences were formed; the patterns of the arrangement were studied; formulas were deduced in each case through the combinatorial principles. The gap software were used to confirmed the total number of elements. Also, the minimum length of factorisation that gives were obtained from the known generating set, for all The findings of the study were: rn • for each the local U-depth of is equal to its defect and global U-depth is . ; rn • the status of satisfies the property ; rn • the number of , , and classes of height r are : , ; ; and respectively. rn • the total number of , , and classes in are: ; ; and respectively. rn • the number of elements in each , and within -class are: ; ; and respectively. rn • the equivalence classes of Green’s relations were also characterized based on their image set and kernel classes. rn In conclusion, some algebraic and combinatorics properties results of subsemigroup OCTn were obtained with relevant examples. This research work and its findings are expected to be beneficial in the areas such as: computational theory, automata theory and formal languages. It can assist in sorting data and designing better network and also, new in.

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