nLab prelattice

1. Idea

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 $K[x]$ of a discrete field $K$, with respect to the divisibility preorder $a \vert b$, the greatest common divisor $\gcd$, and the least common multiple $\lcm$; 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 $K[x]$, where $\mathbb{Z}^\times \coloneqq \{1, -1\}$ and $K[x]^\times \coloneqq K^\times$.

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 unboounded prelattices, or prelattices do not have top and bottom elements, in which those with top and bottom elements are bounded prelattices. A pseudoprelattices is equivalently a thin locally cartesian category whose opposite category is also locally cartesian. Pseudoprelattices are important because given any ordered field $K$ with pseudolattice structure, every ordered Artinian local $K$-algebra, which are used in synthetic differential geometry, is a pseudoprelattice.

2.  See also

• preorder

• total preorder

• Heyting prealgebra

• prelattice-ordered ring

• prelattice object