If one of the binary operations is associative, say is associative, we may define a latticoid as a poset as follows:
Once a latticoid is a poset, we may easily visualize it by a diagram (Hasse diagram), much like that of a lattice. Position above if and connect a line segment between and . The following is the diagram of a latticoid that is meet associative but not join associative:
It is not join associative because , whereas .
Given a latticoid , we can define a dual of by using the same underlying set, and define the meet of and in as the join of and in , and the join of and (in ) as the meet of and in . is a meet-associative latticoid iff is join-associative.
|Date of creation||2013-03-22 16:31:02|
|Last modified on||2013-03-22 16:31:02|
|Last modified by||CWoo (3771)|