symmetric monoidal (∞,1)-category of spectra
A pseudolattice ordered abelian group or l-group is an ordered abelian group whose order forms a pseudolattice.
A pseudolattice ordered abelian group or 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 pseudolattice 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 pseudolattice ordered abelian groups.
An example of a pseudolattice ordered abelian group that is not totally ordered is the abelian group of Gaussian integers with $ramp(1) \coloneqq 1$ and $ramp(i) \coloneqq i$.
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)
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