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 need to 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 ordered abelian group $(G,\le)$ such that every pair of elements $a,b \in G$ admits a meet $a \wedge b$ in the underlying poset (Gilmer 1992, p. 158).
A lattice-ordered abelian group or abelian l-group is an abelian group $G$ with a binary join operation $(-)\vee(-)\colon G \times G \to G$ 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 $ramp(i) \coloneqq i$.
Let $R$ be an integral domain, $F(R)$ its field of fraction and $U(R)$ its group of units. The group of divisibility of $R$ (Gilmer 1992, p.174), denoted $D(R)$, is defined as
The abelian group $D(R)$ becomes an ordered abelian group if we define an order as follows: for every $rU(R),sU(R) \in D(R)$, we say that $rU(R) \le sU(R)$ iff $sr^{-1} \in R$.
[Gilmer 1992, p.174] Let $R$ be an integral domain. Then $D(R)$ is a lattice-ordered abelian group iff $R$ is a GCD domain.
[Gilmer 1992, p.215] Every lattice-ordered abelian group $G$ is order-isomorphic to the group of divisibility of a Bézout domain i.e. there exists a Bézout domain $R$ and a group isomorphism $f:G \rightarrow D(R)$ such that $g \le h \Leftrightarrow f(g) \le f(h)$.
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)
Robert Gilmer: Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics (1992)
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