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Free Bands of Abelian Groups
Author(s) -
Pastijn F. J.,
Trotter P. G.
Publication year - 1991
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-63.2.344
Subject(s) - mathematics , abelian group , rank of an abelian group , elementary abelian group , subvariety , combinatorics , abelian category , abelian variety , free abelian group , idempotence , pure mathematics , free product , discrete mathematics , variety (cybernetics) , group (periodic table) , statistics , chemistry , organic chemistry
If V is a subvariety of the variety A ∘ B of all bands of abelian groups then the free object F V x of V on a countably infinite set X is a subdirect product of the greatest group‐homomorphic image of F V x and the core C ( F V x ) of F V x . For several such varieties V we give a structure theorem for C ( F V x ) that is based on a modification of Nambooripad's description of idempotent‐generated regular semigroups via biordered sets. Using the structure theorem, we show that the word problem for F V x is solvable for these varieties V . Furthermore, we provide a Rees matrix representation for each D‐class in the core of the free band of abelian groups F ( A ∘ B ) X . This enables us to show that the maximal subgroups of F ( A ∘ B ) X are free abelian groups of finite rank and that F ( A ∘ B ) X is residually finite. The lattice of varieties of abelian groups is embeddable in the sublattice of all varieties of bands of abelian groups that contain the variety of all bands. By way of application we determine the lattice of varieties of bands of abelian groups that are given by identities in up to three variables.