We consider particular compatible orders on a given completely simple semi- group Sx= M（（x）; I, A; P） where （x） is an ordered cyclic group with x 〉 1 and p11= x-1. Of these, only the lexicographic and bootlace ...We consider particular compatible orders on a given completely simple semi- group Sx= M（（x）; I, A; P） where （x） is an ordered cyclic group with x 〉 1 and p11= x-1. Of these, only the lexicographic and bootlace orders yield residuated semigroups. With the lexicographic order, Sx is orthodox and has a biggest idempotent. With the bootlace order, the maximal idempotents of Sx are identified by specific locations in the sandwich matrix. In the orthodox case there is also a biggest idempotent and, for sandwich matrices of a given size, uniqueness up to ordered semigroup isomorphism is established.展开更多
The inclusion ideal graph In(S)of a semigroup S is an undirected simple graph whose vertices are all the nontrivial left ideals of S and two distinct left ideals I,J are adjacent if and only if either I⊂J or J⊂I.The p...The inclusion ideal graph In(S)of a semigroup S is an undirected simple graph whose vertices are all the nontrivial left ideals of S and two distinct left ideals I,J are adjacent if and only if either I⊂J or J⊂I.The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of In(S).We investigate the connectedness of In(S)and show that the diameter of In(S)is at most 3 if it is connected.We also obtain a necessary and sufficient condition of S such that the clique number of In(S)is the number of minimal left ideals of S.Further,various graph invariants of In(S),viz.perfectness,planarity,girth,etc.,are discussed.For a completely simple semigroup S,we investigate properties of In(S)including its independence number and matching number.Finally,we obtain the automorphism group of In(S).展开更多
文摘We consider particular compatible orders on a given completely simple semi- group Sx= M（（x）; I, A; P） where （x） is an ordered cyclic group with x 〉 1 and p11= x-1. Of these, only the lexicographic and bootlace orders yield residuated semigroups. With the lexicographic order, Sx is orthodox and has a biggest idempotent. With the bootlace order, the maximal idempotents of Sx are identified by specific locations in the sandwich matrix. In the orthodox case there is also a biggest idempotent and, for sandwich matrices of a given size, uniqueness up to ordered semigroup isomorphism is established.
文摘The inclusion ideal graph In(S)of a semigroup S is an undirected simple graph whose vertices are all the nontrivial left ideals of S and two distinct left ideals I,J are adjacent if and only if either I⊂J or J⊂I.The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of In(S).We investigate the connectedness of In(S)and show that the diameter of In(S)is at most 3 if it is connected.We also obtain a necessary and sufficient condition of S such that the clique number of In(S)is the number of minimal left ideals of S.Further,various graph invariants of In(S),viz.perfectness,planarity,girth,etc.,are discussed.For a completely simple semigroup S,we investigate properties of In(S)including its independence number and matching number.Finally,we obtain the automorphism group of In(S).
