nLab prelattice-ordered commutative monoid

Idea

A prelattice-ordered commutative monoid is a commutative monoid which is also a preordered set with binary meets and binary joins such that the meets and joins preserve the monoid’s binary operation, up to associate elements?.

Definition

Definition

Let $(X, \cdot, 1)$ be a commutative monoid which is also a preordered set $(X, \le)$ and possesses binary meets and binary joins. We say that $X$ is a prelattice-ordered commutative monoid if:

1. For every $x,y,z \in X$, $(x z \wedge y z) \sim (x \wedge y)z$

2. For every $x,y,z \in X$, $(x z \vee y z) \sim (x \vee y)z$

where $\sim$ is the equivalence relation generated by the preorder $\le$.

Example

Let $R$ be a GCD domain. The underlying multiplicative monoid of $R$, together with the divisibility relation is a prelattice-ordered commutative monoid. This is the case because:

1. For every $x,y,z \in R$, $gcd(x z,y z) \sim gcd(x,y)z$

2. For every $x,y,z \in R$, $lcm(x z,y z) \sim lcm(x,y)z$

where $x \sim y$ means that $x$ and $y$ are equal up to multiplication by a unit.

Related concepts

• GCD domain