Category:Infima

From ProofWiki
Jump to: navigation, search

This category contains results about Infima in the context of Order Theory.
Definitions specific to this category can be found in Definitions/Infima.


Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T \subseteq S$.


An element $c \in S$ is the infimum of $T$ in $S$ if and only if:

$(1): \quad c$ is a lower bound of $T$ in $S$
$(2): \quad d \preceq c$ for all lower bounds $d$ of $T$ in $S$.


The infimum of $T$ is denoted $\inf T$.


If there exists an infimum of $T$ (in $S$), we say that $T$ admits an infimum (in $S$).