| Abstract:
Lattices can be viewed as algebras or as ordered sets. In fact, it is
the concept of order that gives a form to otherwise formless lattice
algebra. One cannot assign a meaningful picture for a finite group or a
finite ring but the full meaning of a finite lattice can be faithfully
conveyed by its Hasse diagram. There are many generalizations of lattices
known from the algebra point of view. In this talk, we explore one natural
generalization of lattices from the point of view of partial order and its
basic connection to the algebra: the compatibility of the order relation
with that of algebraic operations. We prove that all well-known lattice
implications (including the equivalence of several forms of distributivity
or modularity etc.) are derivable in this new generalized context as well.
References:
- Lakser, H.; Padmanabhan, R.; Platt, C. R., Subdirect decomposition
of Plonka sums. Duke Math. J. 39 (1972), 485–488.
- Padmanabhan, R.; Penner, P. Structures of free n-quasilattices.
Algebra Colloq. 6 (1999), no. 3, 249–260.
- Padmanabhan, R. Regular identities in lattices. Trans. Amer.
Math. Soc. 158 (1971) 179–188.
- J. Plonka, On a method of construction of abstract algebras, Fund.
Math. 61 (1967), 183–189.
|
