Matrix semigroups over a semigiring

Abstract

If S is an additively commutative semiring and n is a positive integer, let Mn(S) denote the set of all nxn matrices over S, so Mn(S) is a semigroup under matrix multiplication. If S is a commutative semiring and n is a positive integer, for A ɛ Mn(S), the positive determinant of A, det⁺A +, and the negative determinant of A, det⁻A, are defined by det⁺A = [Notation] respectively, where An is the set of all even permutations on {1,2,…,n} and Bn is the set of all odd permutations on {1,2,…,n}. A semigroup S is said to be regular if for every a ɛ S, A = axa for some x ɛ S. A semiring (S,+,•) is called a regular semiring if (S,+) and (S, •) are regular semigroups. A semigroup S is called a semilattice if S is communitative and a² = a for every a ɛ S. A semiring (S,+,•) is called a semilattice semiring if (S,+) and (S, •) are semilattices. In this thesis, we characterize invertible matrices and regular matrix semigroups over semirings with some special properties. Also, three maximal commutative subsemigroups, in explicit forms, of the matrix semigroup Mn(S) with S a commutative semiring with 0, 1 and n a positive integer are introduced. The main results are as follow : Theorem 1. Let S be a commutative semiring with 0, 1. Assume that S has no zero divisors and 0 is the only element of S which has an additive inverse. Then a square matrix A over S is invertible if and only if every row and every column of A has exactly one nonzero element and every nonzero element of A is an invertible element of S. Theorem 2. Let S be a semilattice semiring with 0, 1 and A a square matrix ove S. Then A is invertible if and only if det⁺A + det⁻A = 1 and the product of any two elements of A in the same coloum (row) is 0. Theorem 3. Let S be a semilattice semiring with 0, 1 and A a square matrix over S. Then A is invertible if and only if the product of any two elements of A in the same column [row] is 0 and the sum of all elements of A in each row [column] is 1. Theorem 4. Let S be an additively commutative semiring with 0, n a positive integer and n≥ 3. Then the matrix semigroup Mn(S) is regular if and only if S is a regular ring. Theorem 5. Let S be an additively commutative semiring with 0 and assume that 0 is the only additive idempotent of S. Then for n≥ 2, the matrix semigroup Mn(S) is regular if and only if S is a regular ring. Theorem 6. Let S be a semilattice semiring with 0,1, n a positive integer and n≥ 2. Then the matrix semigroup Mn(S) is regular if and only if n = 2 and S is a Boolean algebra. Theorem 7. Let S be a commutative semiring with 0, 1 and n a positive integer. Then the set of all nxn matrices over S in the form [table] The set of all nxn matrices over S in the form [table] and the set of all nxn matrices over S in the form [table] are maximal commutative subsemigroups of the matrix semigroup Mn(S).