nLab prelattice



Prelattices are lattices which do not satisfy antisymmetry. Or equivalently, they are the bicartesian monoidal preorders, thin cartesian monoidal categories which are also cocartesian monoidal categories.

One example of prelattices include Heyting prealgebras. Two other examples are the integers \mathbb{Z} and the polynomial ring K[x]K[x] of a discrete field KK, with respect to the divisibility preorder a|ba \vert b, the greatest common divisor gcd\gcd, and the least common multiple lcm\lcm; unlike the natural numbers, these only form a prelattice because there are more than one element in the group of units of both \mathbb{Z} and K[x]K[x], where ×{1,1}\mathbb{Z}^\times \coloneqq \{1, -1\} and K[x] ×K ×K[x]^\times \coloneqq K^\times.

In the same way as lattices, one could either assume that prelattices have top and bottom elements, in which those without top and bottom elements are pseudoprelattices, or prelattices do not have top and bottom elements, in which those with top and bottom elements are bounded prelattices. Pseudoprelattices are important because given any ordered field KK with pseudolattice structure, every ordered Artinian local K K -algebra, found in some approaches to analysis, is a pseudoprelattice.

 See also

Last revised on January 23, 2023 at 16:38:20. See the history of this page for a list of all contributions to it.