symmetric monoidal (∞,1)-category of spectra
A lattice-ordered abelian group or l-group is an ordered abelian group whose order forms a lattice. Here, we assume that lattices do not have top or bottom elements, because otherwise the only such object is the trivial group.
A lattice-ordered abelian group or abelian l-group is an abelian group $G$ with a binary join operation $(-)\vee(-):G \times G \to G$ such that $(G, \vee)$ is a commutative idempotent semigroup, and
The meet is defined as
the ramp function is defined as
and the absolute value is defined as
The order relation is defined as in all pseudolattices: $a \leq b$ if $a = a \wedge b$.
The following algebraic definition is from Peter Freyd:
A lattice-ordered abelian group or l-group is an abelian group $G$ with a function $ramp:G \to G$ such that for all $a$ and $b$ in $G$,
and
The join $(-)\vee(-):G \times G \to G$ is defined as
the meet $(-)\wedge(-):G \times G \to G$ is defined as
and the absolute value is defined as
The order relation is defined as $a \leq b$ if $ramp(a - b) = 0$.
All totally ordered abelian groups, such as the integers, the rational numbers, and the real numbers, are lattice-ordered abelian groups.
An example of a lattice-ordered abelian group that is not totally ordered is the abelian group of Gaussian integers with $ramp(1) \coloneqq 1$ and $ramp(i) \coloneqq i$.
prelattice-ordered abelian group?
Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)
Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)
Last revised on February 23, 2024 at 19:59:33. See the history of this page for a list of all contributions to it.