Abstract
As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Mal’cev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.
Highlights
There are various notions of symmetry in the sciences and in mathematics
For a universal algebra the automorphism group/endomorphism monoid consists of all those permutations/selfmaps of the carrier set that commute with all fundamental operations of the algebra
Centralising monoids of single unary operations, i.e., monounary algebras, were investigated in [16,17,18], showing, for example, which centralising monoids of this type are equal to the centralising monoid they describe as a witness [16] (Theorem 4.1, p. 8, Theorem 5.1, p. 10), and which of them have a unique unary operation as their witness [18] (Theorems 3.1 and 3.3, p. 4659 et seq.)
Summary
There are various notions of symmetry in the sciences and in mathematics. Algebraic structures are usually considered symmetric if they have a lot of automorphisms or, more generally, endomorphisms. For a universal algebra (a structure without relations) the automorphism group/endomorphism monoid consists of all those permutations/selfmaps of the carrier set that commute with all fundamental operations of the algebra. Centralising monoids of single unary operations, i.e., monounary algebras, were investigated in [16,17,18], showing, for example, which centralising monoids of this type are equal to the centralising monoid they describe as a witness [16] With respect to unary operations, we establish that on every carrier set of size at least four, every single transposition or every product of disjoint transpositions without fixed points and every non-identical retraction witnesses a centralising monoid which is a co-atom in the lattice of those centralising monoids witnessed by sets of unary operations, that is, in the lattice of endomorphism monoids of unary algebras. We use the same technique to prove that—with the exception of the two-element set—the centralising monoid of a Mal’cev operation of a Boolean group is always maximal
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