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Green's Equivalences and Related Concepts for Semigroups of Continuous Selfmaps
Author(s) -
MAGILL K. D.
Publication year - 1993
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1993.tb52527.x
Subject(s) - mathematics , semigroup , ideal (ethics) , complement (music) , pure mathematics , minimal ideal , prime ideal , identity (music) , prime (order theory) , discrete mathematics , cancellative semigroup , endomorphism , maximal ideal , combinatorics , philosophy , biochemistry , chemistry , physics , epistemology , complementation , acoustics , gene , phenotype
. Green's equivalences are characterized for regular elements of general transformation semigroups and the largest proper left ideal, the largest proper right ideal, and the largest proper two‐sided ideal are all explicitly described when the semigroup has a right identity, a left identity, and a two‐sided identity, respectively. It is shown that the largest proper left ideal and the largest proper right ideal (if they exist) of any semigroup are prime and that the largest proper ideal may, or may not be prime. These results have applications to such semigroups as the semigroup of all endomorphisms of a vector space as well the semigroup, S ( X ), of all continuous selfmaps of a topological space X . However, the semigroup S ( X ) is our primary focus here. For example, it is shown that if X is any local dendrite with finite branch number that is not an arc, then Mon( S ( X )), the semigroup of monomorphisms from S ( X ) into S ( X ), is isomorphic to the dual of the complement of the largest proper right ideal of S ( X ). Furthermore, the complement of the largest proper left ideal of S ( X ) is a homomorphic image of Mon ( S ( X )). Some results are then discussed concerning Green's equivalences for polynomial functions (many of which are irregular) in the semigroup of all continuous selfmaps of the space of real numbers. Finally, the partially ordered families L ( S ( X )) and. R ( S ( X )) are discussed where L ( S ( X )) consists of the equivalence classes of S ( X ) induced by Green's. L ‐equivalence and. R ( S ( X )) consists of the equivalence classes of S ( X ) induced by Green's R ‐equivalence. For example, if X and Y are any two compact 0‐dimensional spaces, then the following four statements are equivalent: (1) L ( S ( X )) and L ( S ( Y )) are order isomorphic, (2) R ( S ( X )) and. R ( S ( Y )) are order isomorphic, (3) the semigroups S ( X ) and S ( Y ) are isomorphic, (4) the topological spaces X and Y are homeomorphic.