Abstract
Let $X_{n}$ be a finite set, $CT_{n}$ the semigroup of full contraction and $OCT_{n}$ order-preserving full contraction transformation semigroup of a finite set. In this paper the Local and global U-depth as well as the status of $OCT_{n}$ were investigated where U is the generating set. The local and global depth were found from the known generating Set of $OCT_{n}$ and also, the status of the semigroup $OCT_{n}$ was obtained from the product of global depth and the order of generating of $OCT_{n}$. For $\alpha\in OCT_{n}$, local depth of α is equal to its defect, global depth and status of $OCT_{n}$ are $ n-1$ and $2(n-1)^{2}$ respectively. We also look at the structure of Green’s relations of order-preserving full contraction transformation semigroup.
Highlights
The algebraic theory of semigroup has been well studied during the second half of the twentieth century
The local and global depth were found from the known generating Set of OCTn and the status of the semigroup OCTn was obtained from the product of global depth and the order of generating of OCTn
We look at the structure of Greens relations of order-preserving full contraction transformation semigroup
Summary
The algebraic theory of semigroup has been well studied during the second half of the twentieth century. Transformation semigroups are one of the most fundamental mathematical objects They occur in theoretical computer science, where properties of language depend on algebraic properties of various transformation semigroups related to them. On studying a new class of semigroup it is mostly of interest to know the type of semigroup in question for example , whether the semigroup is regular, inverse (see Clifford and Preston, 1967). Another interest is in the characterisation of the five classical Green’s relations, mostly when the semigroup is regular. Garba et al(2017) characterised the Green’s relations and starred Green’s relations in CTn, CIn and starred Green’s relation in OCTn
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