Set of Submagmas of Magma under Subset Relation forms Complete Lattice
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Theorem
Let $\struct {A, \odot}$ be a magma.
Let $\SS$ be the set of submagmas of $A$.
Then:
- the ordered set $\struct {\SS, \subseteq}$ is a complete lattice
where for every subset $\TT$ of $\SS$:
- the infimum of $\TT$ necessarily admitted by $\TT$ is $\bigcap \TT$.
Proof
From Magma is Submagma of Itself:
- $\struct {A, \odot} \in \SS$
Let $\TT$ be a non-empty subset of $\SS$.
From Intersection of Submagmas is Largest Submagma:
- $\bigcap \TT \in \SS$
Hence, from Set of Subsets which contains Set and Intersection of Subsets is Complete Lattice:
- $\struct {\SS, \subseteq}$ is a complete lattice
where $\bigcap \TT$ is the infimum of $\TT$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.11 \ \text{(c) (1)}$