Narrow Results

Let be the semigroup of all partial transformations on X, and be the subsemigroups of of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let , and . In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green's relations of . In this paper, we present analogous results for both and . For a finite set X with |X| ≥ 3, the ranks of , and are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of , and for any proper non-empty subset Y of X.

For n ∈ ℕ, let O_n be the semigroup of all singular order-preserving mappings on [n]= {1,2,…,n}. For each nonempty subset A of [n], let O_n(A) = {α ∈ O_n: (∀ k ∈A) kα ≤ k} be the semigroup of all order-preserving and A-decreasing mappings on [n]. In this paper it is shown that O_n(A) is an abundant semigroup with n-1𝒟*-classes. Moreover, O_n(A) is idempotent-generated and its idempotent rank is 2n-2- |A\ {n}|. Further, it is shown that the rank of O_n(A) is equal to n-1 if 1 ∈ A, and it is equal to n otherwise.

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations $A_k{X}_k+Y_k{B}_k=M_k,C_k{X}_{k+1}+Y_k{D}_k=N_k(k=1,2)$ over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

In this short note, we study the rank of a restricted Lie algebra (𝔤, [p]), and give some applications which concern the dimensions of non-trivial irreducible modules. We also compute the rank of the restricted contact algebra.