let tx be the full transformation semigroups on the set x. for an equivalence e on x, let te(x) = {α ∈ tx : ∀(x, y) ∈ e ⇔ (xα, yα) ∈ e}it is known that te(x) is a subsemigroup of tx. in this paper, we discussthe green's *-relations, certain *-ideal and certain rees quotient semigroup for te(x).

this text carries out some ideas about exact and p− exact sequences of transformationsemigroups. some theorems like the short five lemma (lemma 1.3 and lemma 2.3) are valid here as inexact sequences of r − modules.

in the transformation semigroup (x, s) we introduce the height of a closed nonempty invariant subset of x, define the transformed dimension of nonempty subset s of x and obtain some results and relations.

The full order-preserving transformation semigroup, the orderpreserving partial transformation semigroup and the order-preserving one-to-one partial transformation semigroup on a poset X are denoted by OT (X), OP (X) and OI(X), respectively. It is well-known that for any posets X and Y , OT (X) ∼= OT (Y ) if and only if X and Y are either order-isomorphic or order-anti-isomorphic. The purpose o...

In this paper we propose a general recipe for calculating the automorphism groups of semigroups consisting of partial endomorphisms of relational structures with a single m-ary relation for any m ∈ N over a finite set. We use this recipe to determine the automorphism groups of the following semigroups: the full transformation semigroup, the partial transformation semigroup, and the symmetric in...

Tetris is a popular puzzle video game, invented in 1984. We formulate two versions of the game as transformation semigroup and use this formulation to view through lens Krohn-Rhodes theory. In variation upon which it restarts if player loses, we find permutation group structures, including symmetric \(S_5\) contains non-abelian simple subgroup. This implies, at least case, that iterated finitar...

Let TX be the full transformation semigroups on the set X. For an equivalence E on X, let TE(X) = {α ∈ TX : ∀(x, y) ∈ E ⇔ (xα, yα) ∈ E}It is known that TE(X) is a subsemigroup of TX. In this paper, we discussthe Green's *-relations, certain *-ideal and certain Rees quotient semigroup for TE(X).

This paper analyses the construction of the kernel graph of a non-synchronizing transformation semigroup and introduces the inverse synchronization problem. Given a transformation semigroup S ≤ Tn, we construct the kernel graph Gr(S) by saying v and w are adjacent, if there is no f ∈ S with vf = wf . The kernel graph is trivial or complete if the semigroup is a synchronizing semigroup or a perm...