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This book addresses Birkhoff and Mal'cev's problem of describing subquasivariety lattices. The text begins by developing the basics of atomic theories and implicational theories in languages that may, or may not, contain equality. Subquasivariety lattices are represented as lattices of closed algebraic subsets of a lattice with operators, which yields new restrictions on the equaclosure operator. As an application of this new approach, it is shown that completely distributive lattices with a dually compact least element are subquasivariety lattices. The book contains many examples to illustrate these principles, as well as open problems. Ultimately this new approach gives readers a set of tools to investigate classes of lattices that can be represented as subquasivariety lattices.
List of contents
Preface.- Introduction.- Varieties and quasivarieties in general languages.- Equaclosure operators.- Preclops on finite lattices.- Finite lattices as Sub(S, , 1, ): The case J(L) (L).- Finite lattices as Sub(S, , 1, ): The case J(L) (L).- The six-step program: From (L, ) to (Lq( ), ).- Lattices 1 + L as Lq( ).- Representing distributive dually algebraic lattices.- Problems and an advertisement.- Appendices.
Report
"This is a research monograph that reports on investigations, both classical and new, into the structure of subquasivariety lattices. ... This monograph is a study of the structure of lattices of the form Lq(K). In this monograph, relation symbols are permitted. ... The monograph spans 290 pages and has 10 chapters and three appendices." (Keith A. Kearnes, Mathematical Reviews, April, 2024)
