Semigroup Of Binary Reltions And Its Generalization

In this paper,two classes of semigroups of binary relations are introduced:a class is a semigroup P_θ(I×∧)of binary relations from a set I to a set∧,that is a generalization of semigroup P(∧×∧)of all binary relations and sandwish semigroup P_θ(∧×∧)of binary relations on a set∧;the other class is a semigroup P_Γ(∧×∧)of binary relations determined by a semilatticeΓon a set∧,that is a subsemigroup of semigroup P(∧×∧)of all binary relations on a set∧,and is a generalization of semigroup of all rectangular binary relations on a set∧.Let I,∧be two arbitrary nonempty sets,and let P(I×∧)be the set of all binary relations from the set I to the set∧.Choose and fix an elementθof the set P(∧×I),andθ≠φ,then the set P(I×∧)is a semigroup under the binary given byρ·σ=ρoθoσ={(i,λ)∈I×Λ;(彐(μ,j)∈θ)(i,μ)∈ρand(j,λ)∈σ}, and is called a semigroup of binary relations from the set I to the set∧,and is denoted by P_θ(I×∧).The paper finds a Boolean matrix representation of the semigroup P_θ(I×∧);researches specific elements of the semigroup P_θ(I×∧)and some properties of Green's relations.Whenθ=∧'×I(?)∧×I,the structure of idempotent elements of the semigroup P_θ(I×∧) are researched;shows that the semigroup P_θ(I×∧)is a completely qusi-regular semigroup; non-solvable elements and the irreducible generating sets of the semigroup P_θ(I×∧)are established;the Green's relations of the semigroup P_θ(I×∧)are obtained;some specific count of the semigroup P_θ(I×∧)are discussed;in the end,whenθ=∧×I'(?)∧×I,some similar properties of the semigroup P_θ(I×∧)are analogously listed.Let∧be an arbitrary nonempty set.let P(∧)be the power set of the set∧,and be a join semilattic in the set theory.LetΓbe a subsemilattice of semilattice P(∧),then it is called as a semilattice on the set∧.Let f:∧→Γbe a set-valued mapping,definesα_f=(?)λf×{λ},obviouslyα_f∈P(∧×∧).DenotesP_Γ(∧×∧)={α_f:f be a set-valued mapping of the set A to the setΓ}, then P_Γ(∧×∧)is a subsemigroup of the semigroup P(∧×∧)of all binary relations on the set∧,and is called a semigroup P_Γ(∧×∧)of binary relations determined by the semilattice Γon the set∧.In the paper,some properties of Green's R-relations,non-solvable elements,Green's(?)-relations and idempotent elements of the semigroup P_Γ(∧×∧)are discussed.Next,the paper introduces a concept of simple semilattice on the set∧.LetΓbe a semilattice on the set∧, callΓis simple,ifΓsatisfies the following two conditions:(1)(?)U,V∈Γ,and U≠V,then U∪V=sup(Γ);(2)(?)U,V∈Γhave U∩V≠φ. WhenΓis a simple semilattice on the set∧,the paper establishes the structure of idempotent elements and maximal subgroups of the semigroup P_Γ(∧×∧);shows that the semigroup P_Γ(∧×∧)is a completely qusi-regular semigroup;finds the irreducible generating sets of the semigroup P_Γ(∧×∧);researches the structure of regular elements of the semigroup P_Γ(∧×∧); finally,discusses some specific count of the semigroup P_Γ(∧×∧).