By Olexandr Ganyushkin

The target of this monograph is to provide a self-contained advent to the fashionable conception of finite transformation semigroups with a robust emphasis on concrete examples and combinatorial purposes. It covers the next issues at the examples of the 3 classical finite transformation semigroups: variations and semigroups, beliefs and Green's family members, subsemigroups, congruences, endomorphisms, nilpotent subsemigroups, displays, activities on units, linear representations, cross-sections and variations. The booklet comprises many workouts and ancient reviews and is directed, to start with, to either graduate and postgraduate scholars searching for an advent to the idea of transformation semigroups, yet must also turn out worthy to tutors and researchers.

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2. 4 each principal left ideal of IS n is uniquely determined by the set dom(α), which is just a subset of N. Hence the number of such ideals is 2n . 9. 7 Let S denote one of the semigroups Tn , PT n , or IS n , and α ∈ S be an element of rank k. , S = IS n . ⎪ ⎩ i i i=0 Proof. We start with the cases S = Tn and S = PT n . As πα ⊂ πβ , each transformation β ∈ Sα is uniquely determined by its values on the set of those equivalence classes of πα which are contained in dom(α). We have k such classes and for each of them we have to choose the value of β on this class, which is an element from N (or N ∪ {∅} in the case of PT n ).

4 Let α, β ∈ PT n . Then the element β is inverse to the element α if and only if the following conditions are satisﬁed: (a) dom(β) ⊃ im(α). (b) β(a) ∈ {x ∈ N : α(x) = a} for all a ∈ im(α). (c) im(β) = β(im(α)). Proof. Let im(α) = {a1 , a2 , . . , ak }. For i = 1, . . , k set Bi = {x ∈ N : α(x) = ai }. 5) β(ai ) ∈ Bi for alli = 1, 2, . . , k. 5) is satisﬁed. Then for all i we have (αβ)(ai ) = ai and (βαβ)(ai ) = β(ai ). Set B = {β(y) : y ∈ im(α)}. 6. REGULAR AND INVERSE ELEMENTS 25 the previous equality we have (βα)(b) = b for all b ∈ B.

7 The notation for the elements of IS n , introduced in Sect. 5, is close to the notation used in [Li]. 8 The role of IS n in the theory of inverse semigroups is analogous to the role of Sn in group theory, and to the role of Tn in semigroup theory. 5 (Preston–Wagner) Any inverse semigroup T is isomorphic to a subsemigroup of the semigroup IS(T ) of all partial injective transformations of the set T . The idea of the proof is as follows: for each a ∈ T we consider the transformation ρa : a−1 T → aT , deﬁned via ρa (x) = ax.