We consider the lattice of pseudovarieties contained in a given pseudovariety P. It is shown that if the lattice L of subpseudovarieties of P has finite height, then L is isomorphic to the lattice of subvarieties of a locally finite variety. Thus not every finite lattice is isomorphic to a lattice of subpseudovarieties. Moreover, the lattice of subpseudovarieties of P satisfies every positive universal sentence holding in all the lattices of subvarieties of varieties V (A) generated by algebras A âP.
Agliano', P., Nation, J.B. (1989). Lattices of pseudovarieties. JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES A-PURE MATHEMATICS AND STATISTICS, 46(2), 177-183 [10.1017/S1446788700030640].
Lattices of pseudovarieties
AGLIANO', PAOLO;
1989-01-01
Abstract
We consider the lattice of pseudovarieties contained in a given pseudovariety P. It is shown that if the lattice L of subpseudovarieties of P has finite height, then L is isomorphic to the lattice of subvarieties of a locally finite variety. Thus not every finite lattice is isomorphic to a lattice of subpseudovarieties. Moreover, the lattice of subpseudovarieties of P satisfies every positive universal sentence holding in all the lattices of subvarieties of varieties V (A) generated by algebras A âP.
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https://hdl.handle.net/11365/1014142