The structure of the clone lattice and Galois connection in multiple-valued logic

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For a set A, a clone on A is a set of multi-variable functions on A which is closed under composition. The set of all clones on A forms the lattice. We also consider the lattice of all monoids consisting of unary functions, In this research, we considered a naturally defined Galois connection between both lattices. For a monoid M, the centralizer of M is the set of all multi-variable functions which ‘commutes' with all unary functions in M.1.Some fundamental properties of the Galois connection :(1)We studied roughly the positions of the centralizers of monoids in the lattice of clones. (2)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.3.A sufficient condition for the centralizer of a monoid to be the least clone :We found a sufficient condition for the centralizer of a monoid to be the least clone which can be used as a very convenient tool.4.The centralizers of monoids containing the symmetric group :We determined the centralizers of all monoids which contain the symmetric group. In the course of this research, the above mentioned sufficient condition has been used quite effectively. For most monoids, their centralizers turned out to be the least clone. However, the monoid "M_2" defined over the 4 element set is an exception and its centralizer is not the least clone.5.Monoids whose centralizer is the least clone :It is ‘natural' to think that under a Galois connection a small monoid corresponds to a large monoid. However, against this intuition, some small monoids have been discovered whose centralizer is the least clone.