Article,
An equational theory for trilattices
algebra universalis, 42 (4): 253--268 (1999)
Abstract
This paper presents the new algebra of trilattices, which are
understood as the triadic generalization of lattices. As with
lattices, there is an order-theoretic and an algebraic approach to
trilattices. Order-theoretically, a trilattice is defined as a
triordered set in which six triadic operations of some small arity
exist. The Reduction Theorem guarantees that then also all finitary
operations exist in trilattices. Algebraically, trilattices can be
characterized by nine types of trilattice equations. Apart from the
idempotent, associative, and commutative laws, further types of
identities are needed such as bounds and limits laws, antiordinal,
absorption, and separation laws. The similarities and differences
between ordered and triordered sets, lattices and trilattices are
discussed and illustrated by examples.
