Definition:Infimum/Ordered Set
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Definition
Let $\left({S, \preceq}\right)$ be a poset.
Let $T \subseteq S$.
An element $c \in S$ is the infimum of $T$ in $S$ 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 $\inf \left({T}\right)$.
The infimum of $x_1, x_2, \ldots, x_n$ is denoted $\inf \left\{{x_1, x_2, \ldots, x_n}\right\}$.
If there exists an infimum of $T$ (in $S$), we say that $T$ admits an infimum (in $S$).
The infimum of $T$ is often called the greatest lower bound of $T$ and denoted $\operatorname{glb} \left({T}\right)$.
Also defined as
Some sources refer to the infimum as being the lower bound. Using this convention, any number smaller than this is not considered to be a lower bound.
Linguistic Note
The plural of infimum is infima, although the (incorrect) form infimums can occasionally be found if you look hard enough.
Also see
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 14$: Order
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definition $1.1.2$
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 2.6$
