Definition:Operator Generated by Ordered Subset
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Definition
Let $L = \struct {X, \precsim}$ be an ordered set.
Let $S = \struct {T, \preceq}$ be an ordered subset of $L$.
The operator generated by ordered set $S$ is defined by
- $\forall x \in X: \map {\map {\operatorname {operator} } S} x := \map {\inf_L} {x^\succeq \cap T}$
where $x^\succeq$ denotes the upper closure of $x$.
Sources
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- Mizar article WAYBEL10:def 5