An equational theory for trilattices

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.