Galois connection in mathematical clone theory

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For a set A, a clone on A is a set of multi-variable functions which is closed under composition. Denote by L_A the set of all clones on A. In this reseach, for the set M_A of all monoids consisting of unary functions on A, we considered a naturally defined Galois connection between M_A and L_A. For a monoid M, the centralizer M^* of M is the set of all multi-variable functions which 'commutes' with every unary function in M.1. <Some fundamental properties of the Galois connection> : (i) We showed that all centralizers of monoids are contained in some particular maximal clones. (ii) Also, we showed that for every pair of distinct monoids their centralizers are always distinct.2. <Characterization of the centralizers of the symmetric group and the alternating group> : We established the characterization of the centralizers of both the symmetric group and the alternating group, the latter being more complex than the former.3. <Classification of the centralizers for a sequence of monoids which contain the symmetric group> : A typical sequence {N_i} of monoids containing the symmetric group was defined. The centralizers of all N_i's have been determined. Most of them coincide with the least clone.4. <Monoids whose centralizer is the least done> : It is 'natural' to think that under a Galois connection a small monoid corresponds to a big monoid. However, against this naive intuition, some small monoids have been discovered whose centralizer is the least clone.5. <Application of the Kuznetsov criterion> : The power of the criterion established by Kuznetsov was shown to be quite useful in our study.