Abstract
Let X be a nonempty set and P(X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End1P(X)(P(X)) of unital ring endomorphisms of the Boolean ring P(X) and the automorphism group Aut(P(X)) when X is finite. Among other facts, it is shown that if X has cardinality n≥1, then End1P(X)(P(X))≅Tnop, where Tn is the full transformation monoid on the set X and Aut(P(X))≅Sn.
Highlights
The aim of this paper is to provide an explicit description of the semigroup of unital endomorphisms and the group of automorphisms of a finite Boolean ring
Note that P ( X ) is the typical example of a Boolean ring, with the symmetric difference playing the role of addition and the intersection playing the role of multiplication
It is worth noting that every finite Boolean ring is isomorphic to a power set ring for some set Ω
Summary
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