The Structures And Properties Of Regular Linear Transformation Semigroups

In this paper,we completely obtain the structures and classifications of the maximal regular subsemigroups, the maximal orthodox subsemigroups and the maximal inverse subsemi-groups of the regular linear transformation semigroups R(lT_n). Moreover, we also get the structures and classifications of those sub semigroups of the singular regular linear transformation semigroups Sl_n. The main results are the following theorems:Theorem 1 Let n (i = 1,2,3, ...t)distinct primes. Denote Sl_n = R(lT_n) \ lD_n. Then the maximal regular subsemigroup M of R(lT_n) must be one of the following types:(a) M₁ = Sl_n U G, G is a maximal subgroup of lD_n;(b) M₂ = lD_n U (Sl_n \ ID_ri, r_i = n/q_i, q_i = p_i^ki1 ≤ t ≤ t.Theorem 2 Let n Pi (i = 1,2,3, ...t)distinct primes. Then the maximal regular subsemigroups M of the singular linear transformation semigroup Sl_n = R(lT_n)\lD_n must be one of the following types: where K is a maximal subgroup of a given group H-class H contained in Theorem 3 Let n = rd, H be an H-classe of . DenoteThen the maximal inverse subsemigroup of R(lTn) must be the following types:Theorem 4 Let n = rd, H₁ å’Œ H₂ are H-classes of ID_r, N_o = , thenMoreover, if Pi distinct primes, then, up to isomorphism, R(lT_n) has 2^t maximal inverse subsemigroups .