Let λ = (λ 1 , λ 2 , . . .) be a partition of n, a sequence of positive integers in non-increasing order with sum n. Let Ω := {1, . . . , n}. An ordered partition P = (A 1 , A 2 , . . .) of Ω has type λ if |A i | = λ i .Following Martin and Sagan, we say that G is λ-transitive if, for any two ordered partitions P = (A 1 , A 2 , . . .) and Q = (B 1 , B 2 , . . .) of Ω of type λ, there exists g ∈ G with A i g = B i for all i. A group G is said to be λhomogeneous if, given two ordered partitions P and Q as above, inducing the setsThe first goal of this paper is to classify the λ-homogeneous groups (Theorems 1.1 and 1.2). The second goal is to apply this classification to a problem in semigroup theory.Let Tn and Sn denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H Sn. Given a non-invertible transformation a ∈ Tn \ Sn and a group G Sn, we say that (a, G) is an H-pair if the semigroups generated by {a} ∪ H and {a} ∪ G contain the same non-units, that is, a, G \ G = a, H \ H. Using the classification of the λ-homogeneous groups we classify all the Sn-pairs (Theorem 1.7). For a multitude of transformation semigroups this theorem immediately implies a description of their automorphisms, congruences, generators and other relevant properties (Theorem 8.5).This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas.The paper finishes with a number of open problems on permutation and linear groups.

Denote by T (X) the semigroup of full transformations on a set X. For ε ∈ T (X), the centralizer of ε is a subsemigroup of T (X) defined by C(ε) = {α ∈ T (X) : αε = εα}. It is well known that C(id X ) = T (X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(id X ) contains all non-invertible transformations in C(id X ).This paper generalizes this result to C(ε), an arbitrary regular centralizer of an idempotent transformation ε ∈ T (X), by describing the subsemigroup generated by the idempotents of C(ε). As a corollary we obtain that the subsemigroup generated by the idempotents of a regular C(ε) contains all non-invertible transformations in C(ε) if and only if ε is the identity or a constant transformation.

An efficient management of physical assets in general and the urban transportation buses in particular, has as objective the optimization of its Life Cycle Cost (LCC), being the optimization of its maintenance and the determination of the right moment of its withdrawal or its renewal, essential elements for optimization of the capital investments. These aspects are related to the determination of the optimal dimension of the fleet reserve, aiming the maximization of its availability and the investment minimization. This can be reached through several algorithms that ought to be in consideration aspects like the maintenance costs, reseller cost, and inflation, among others. This is reached trough the adequate maintenance policies, being these scheduled and or on-condition predictive or any kind of planned ones. This type of decisions are determinant to maximize the LCC and, by consequence, the dimension of reserve fleet. It is through this perspective that the identification of the right moment of replacement or total renewal of an asset can be a strategic element in the competitiveness of the organizations by reducing the costs that can come from there. The validation of the management models to be implemented will be done in the road transports sector of passengers' transportation and the models and the results have the potential to help this strategic sector in the global economy.

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The symmetric inverse monoid In is the set of all partial permutations of an n-element set. The largest possible size of a 2-generated subsemigroup of In is determined. Examples of semigroups with these sizes are given. Consequently, if M (n) denotes this maximum, it is shown that M (n)/|In| → 1 as n → ∞. Furthermore, we deduce the known fact that In embeds as a local submonoid of an inverse 2-generated subsemigroup of I n+1 .

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