symmetric monoidal (∞,1)-category of spectra
A lattice-ordered ring is an partially ordered ring whose partial order forms a lattice. Lattices here are assumed not to have top or bottom elements, because otherwise the only such lattice-ordered ring is the trivial ring.
A lattice-ordered ring or l-ring is a preordered ring where the partial order $\lt$ is a lattice: it has binary joins and meets.
If the relation $\leq$ is only a preorder, then the preordered ring $R$ is said to be a prelattice-ordered ring.
The following essentially algebraic definition is adapted from the algebraic definition of lattice-ordered abelian group by Peter Freyd:
A lattice-ordered ring is an ring $R$ with a function $ramp:R \to R$ such that for all $a$ and $b$ in $G$,
and the following Horn clause:
An element $a$ in $R$ is non-negative if $ramp(a) = a$. The Horn clause can then be stated as multiplication of non-negative elements is non-negative.
The join $(-)\vee(-):R \times R \to R$ is defined as
the meet $(-)\wedge(-):R \times R \to R$ is defined as
and the absolute value is defined as
The order relation is defined as in all lattices: $a \leq b$ if $a = a \wedge b$.
All totally ordered rings, such as the integers, the rational numbers, and the real numbers, are lattice-ordered rings.
Wikipedia, Lattice-ordered ring
Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)
Last revised on February 23, 2024 at 19:52:00. See the history of this page for a list of all contributions to it.