We say that A is a sup-[OMEGA]-algebra if the poset (A[less than or equal to]) is a complete lattice and

j] is a complete lattice with respect to the order inherited from A, where joins are given by

If A is a complete lattice, M [subset or equal to] A is closed under joins and meets, and [a.

From the algebraic point of view this structure forms a complete lattice, called the generalized one-sided concept lattice.

The structure L of truth degrees forms a so-called complete lattice, i.

The direct product of lattices forms a complete lattice if and only if all members of the family are complete lattices.

We use 1 to denote the top element and 0 the bottom element in a complete lattice.

A quantale is a complete lattice Q with an associative binary operation "&" satisfying:

Any complete lattice implication algebra is a Girard quantale with unique cyclic dualizing element 0.

In the trivial case, if we take the complete lattice [L.

While the modest generalization of allowing [infinity] as a similarity value does not pose a serious restriction (databases are usually built from finite, and therefore bounded sets of data), it makes the set [0, [infinity], when ordered by the usual [greater than or equal to] relation, a complete lattice [L.

A](x,x) = 0, which is the greatest element of the complete lattice [L.