Some Studies Of Principal Factors Of Several Transformation Semigroups

Let n∈N+, Xn = {1,2,..., n} ordered in the standard way. We denote by Pn the semigroup (under composition) of all partial transformations of Xn, and we denote by Jn the semigroup (under composition) of all transformations of Xn. Let Rn={α∈Jn:|xα-1|≥|im(α)|((?)x∈im(α))},PRn={α∈Pn:|xα-1|≥|im(α)|((?)x∈im(α))},It is easy to show that Rn is a subsemigroup of Jn and PRn is a subsemigroup of Pn.We have the following results:In Chapter 2,for arbitrary n > 4, we study the complete classification of maximal regular subsemigroups of principal factors of semigroup Rn, the complete classification of idempotent-generated maximal regular subsemigroups of principal factors of semigroup Rn,and the complete classification of square idempotent-generated maximal regular sub-semigroups of principal factors of semigroup Rn, the following results are given:Theorem 2.1 Let 2≤r≤[(?)]. Then each maximal regular subsemigroup of principal factors Pr of semigroup Rn must be one of the following forms:(1) Mα=(0)∪(Jr\Rα),α∈Jr;(2)Nβ={0}∪(Jr\Lβ),β∈Jr.Theorem 2.2 Let 3 ≤r≤[(?)]. Then the idempotent-gecerated maximal regular and maximal regular subsemigroups of principal factors Pr of semigroup Rn coincide.Theorem 2.3 Let 3≤r≤[(?)]. Then the square idempotent-generated maxi-mal regular and maximal regular subsemigroups of principal factors Tr of semigroup Rn coincide.In Chapter 3, we study the idempotent depth of semigroup PRn, we have the following results:Theorem 3.1 PRn is a regular subsemiband of Pn and △(PRn) = 2 if and only if n>1.Theorem 3.2 Greern relations on PRn are characterized by(ⅰ)αLβ and only if im(α) = im(β)(ⅱ)αRβ if and only if ker(α)=ker(β)(ⅲ)αDβ if and only if |im(α)|=|im(β)|(ⅳ)D=l.In Chapter 4,for arbitrary n≥4, we study the complete classification of maximal regular subsemigroups of principal factors of semigroup PRn, the complete classification of idempotent-generated maximal regular subsemigroups of principal factors of semigroup PRn,and the complete classification of square idempotent-generated maximal regular subsemigroups of principal factors of semigroup PRn, the following results are given:Theorem 4.1 Let 2≤r≤[(?)]. Then each maximal regular subsemigroup of principal factors Pr of semigroup PRn must be one of the following forms:(1) PMα= {0} ∪(Jr\Rα),α∈Jr;(2)PNβ={0}∪(Jr\Lβ),β∈Jr.Theorem 4.2 Let 3 ≤r≤[(?)]. Then the idempotent-generated maximal regular and maximal regular subsemigroups of principal factors Pr of semigroup PRn coincide.Theorem 4.3 Let 3 ≤r≤[(?)]. Then the square idempotent-generated maximal regular and maximal regular subsemigroups of principal factors Pr of semigroup PRn coincide.