Varieties of implicative semi-lattices. II

Varieties of Implicative Semilattices

Varieties of Lattices

In this chapter we discuss some of the more recent results and give a general overview of what is currently known about lattice varieties. Of course it is impossible to give a comprehensive account. Often we only cite recent or survey papers, which themselves have many more references. We would like to apologize in advance for any errors, omissions, or miscrediting of results. For proofs of the...

Varieties of Residuated Lattices

ACKNOWLEDGMENTS This thesis wouldn't have been possible without the help and support of certain people. I could not possibly itemize my gratitude in detail, but I would like to mention some of the most important names. First and foremost, I would like to thank my advisor, Constantine Tsinakis, for introducing me to the subject of residuated lattices, and for his guidance and advice. Despite his...

Minimal varieties of residuated lattices

In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Fi...

Quasi-Implicative Lattices and the Logic of Quantum Mechanics

Mittelstaedt has defined the class of quasi-implicative lattices and shown that an ortholattice (orthocomplemented lattice) is quasi-implicative exactly if it is orthomodular (quasi-modular). He has also shown that the quasi-implication operation is uniquely determined by the quasi-implicative conditions. One of Mittelstaedt's conditions, however, seems to lack immediate intuitive motivation. C...