Let Xn = {1, 2, . . . , n} with its natural order and let Tn be the full transformation semigroup on Xn . A map α ∈ Tn is said to be order-preserving if, for all x, y ∈ Xn , x ≤ y ⇒ xα ≤ yα . The map α ∈ Tn is said to be a contraction if, for all x, y ∈ Xn , |xα − yα| ≤ |x − y| . Let CT n and OCT n denote, respectively, subsemigroups of all contraction maps and all order-preserving contraction maps in Tn . In this paper we present characterisations of Green's relations on CT n and starred Green's relations on both CT n and OCT n .
In this paper, the authors introduced a novel definition based on Hilfer fractional derivative, which name $q$-Hilfer fractional derivative of variable order. And the uniqueness of solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order of the form \eqref{eq:varorderfrac} with $0 < \alpha(t) < 1$, $0 \leq \beta \leq 1$, and $0 < q < 1$ is studied. Moreover, an example is provided to demonstrate the result.
Let Singn denotes the semigroup of all singular self-maps of a finite set Xn={1,2, . . . , n}. A map α∈Singn is called a 3-path if there are i, j, k∈Xn such that iα=j,jα=k and xα=x for all x∈Xn\ {i, j}. In this paper, we described aprocedure to factorise each α∈Singn into a product of 3-paths. The length of each factorisation, that is the number of factors in eachfactorisation, is obtained to be equal to ⌈12(g(α)+m(α))⌉, where g(α) is known as the gravity of α and m(α) is a parameter introduced inthis work and referred to as the measure of α. Moreover, we showed that Singn⊆P[n−1], where P denotes the set of all 3-paths in Singn and P[k]=P ∪P2∪ ··· ∪Pk.
For a non-empty set X denote the full transformation semigroup of X by T(X). Let \(\sigma\) be an equivalence relation on X and E(X, \(\sigma\)) denotes the semigroup (under composition) of all \(\alpha\) : X \(\mapsto\) X, such that \(\sigma\) \(\subseteq\) ker(\(\alpha\) ). Semigroup of transformations with restricted equivalence occur when we take all transformations whose kernel is contained in some fixed equivalence, E(X, \(\sigma\)). First, we found that E(X, \(\sigma\)) is a disjoint union copies of two generating sets. Next, we discuss the presentations, acts, subacts, direct products and bilateral semidirect product of the semigroup of transformation with restricted equivalence E(X, \(\sigma\)) and its application.
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