Some Finiteness Conditions for Strong Semilattice of Semigroups

Let , ; , i j i S I S φ   =   S be a strong semilattice of semigroups such that I is finite and each i S ( ) i I ∈ be a family of disjoint semigroups. In this article some finiteness conditions which are periodicity, local finiteness and locally finite presentability are considered for S . It is proven that a strong semilattice of semigroups , ; , i j i I S φ     S is periodic, locally finite, locally finitely presented and residually finite, respectively if and only if I is finite and each semigroup i S ( ) i I ∈ is periodic, locally finite, locally finitely presented and residually finite, respectively.


Introduction
The study of finiteness conditions for semigroups (the properties of semigroups which all finite semigroups have) is one of the most important topics for the mathematical community. Some of the properties of finiteness are being finitely generated, finite presentability, having a soluble word problem, periodicity, local finiteness, local finite presentability, residual finiteness, being hopfian, having finite complete rewriting systems and finite derivation type (FDT). They have been considered for certain classes of semigroup constructions (see [1][2][3][4][5][6][7][8][9]). In [1], it was shown that the properties of being finite, being finitely generated, being finitely presented and being residually finite are preserved by large subsemigroups and small extensions. In [5], periodicity, local finiteness, residual fniteness of Rees matrix semigroups and solvable word problem for Rees matrix semigroups were investigated.
One of semigroup structures is a strong semilattice of semigroups which is an important structure for completely regular semigroups (see [10]). Finite presentability of strong semilattice of semigroups was investigated in [3]. The authors of [3]  In this paper periodicity, locally finiteness, locally finite presentability and residual finiteness of strong semilattice of semigroups are investigated.

Notation and Basic Definitions
Let P be any property of semigroups. It said that a semigroup is locally P if every finitely generated subsemigroup of has property P. If P is a finiteness condition, then being locally P is also a finiteness condition. In particular, being locally finite and being locally finitely presented are finiteness conditions. Let I be a semilattice, and let i S ( ) i I ∈ be a family of disjoint semigroups. Suppose that, for any two elements , and that these homomorphisms satisfy the following conditions: Define a multiplication on With respect to this multiplication, S is a semigroup, called a strong semilattice of semigroups, and is denoted by , ; , Let A be an alphabet, A + be the free semigroup on A (i.e. the set of all non-empty words over A ). A semigroup presentation is an ordered pair A R where R A A An element a of A is called a generating symbol, while an

Periodicity
Recall that a semigroup S is periodic if, for each s S ∈ , the monogenic semigroup generated by s is finite, or equivalently there exist distinct positive integers m and r (depending on s ) such that m r m s s + = . , is periodic if and only if, for is a subsemigroup of the strong semilattice S and the subsemigroups of the periodic semigroups are periodic, each semigroup i Then, there exists an i such that So the strong semilattice , ; ,

Local Finiteness
Let X be any non-empty subset of a semigroup S . Then the smallest subsemigroup of S containing X is called the subsemigroup of S generated by X , and is denoted by X . If X is finite then X is called a finitely generated subsemigroup. If every finitely generated subsemigroup of S is finite, then S is called locally finite. , , is locally finite.

Locally Finite Presentability
Let S be a semigroup. If every finitely generated subsemigroup of S is finitely presented, then S is called locally finitely presented.
And for all ( ) k P K ∈ , k Z can be defined as follows ve ( )

Residual Finiteness
Residual finiteness is one of the most important algebraic finiteness conditions, and has been widely studied in the context of groups, semigroups, monoids and other algebraic structures. Let X be a set and let π be an equivalence relation on X. For x ∈ X we use x/π to denote the equivalence class of the element x, and X/π to denote the set of all equivalence classes. The number of π-classes of X is denoted [X: π] and called this the index of the relation π. It is said that π separates the elements s and t if s/π = t/π. A semigroup S is residually finite if for every two distinct x, y ∈ S there is a congruence ρ on S which has finite index and which separates x and y. This is equivalent to saying that there is a homomorphism Φ from S onto a finite semigroup such that