# Infimum

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Fleshing out, make sure the caveat is known, proof of claim

## Definition

Let [ilmath](X,\preceq)[/ilmath] be a poset and let [ilmath]A\subseteq X[/ilmath] be any subset of [ilmath]X[/ilmath][Note 1]. The infimum (AKA: greatest lower bound, g.l.b) of [ilmath]A[/ilmath] is an element of [ilmath]X[/ilmath], written [ilmath]\text{Inf}(A)[/ilmath] that satisfies the following two conditions:

1. [ilmath]\forall a\in A[\text{Inf}(A)\preceq a][/ilmath] - which states that [ilmath]\text{Inf}(A)[/ilmath] is a lower bound of [ilmath]A[/ilmath] - and
2. [ilmath]\forall b\in\underbrace{\left\{x\in X\ \vert\ (\forall a\in A[x\preceq a])\right\} }_{\text{the set of all lower bounds of }A }\Big[b\preceq\text{Inf}(A)\Big][/ilmath] - which states that for all lower bounds of [ilmath]A[/ilmath], that lower bound "is majorised by"[Note 2] [ilmath]\text{Inf}(A)[/ilmath]
• Claim 1: we have part 2 of the definition if and only if [ilmath]\forall x\in X\Big[\underbrace{\left(\forall a\in A[x\preceq a]\right)}_{x\text{ is a lower bound of }A}\implies x\preceq\text{Inf}(A)\Big][/ilmath]
• Claim 2: we claim 1 if and only if [ilmath]\left(A=\emptyset\vee\Big(\forall x\in X\exists a\in A[x\succ\text{Inf}(A)\implies a\prec x]\Big)\right)[/ilmath]

Notice the [ilmath]A=\emptyset[/ilmath] condition here, as in the case [ilmath]A[/ilmath] is empty, [ilmath]\exists a\in A[/ilmath] is always false. This is a very big caveat.