Set of Submagmas of Magma under Subset Relation forms Complete Lattice

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 $\ds \bigcap \TT$.

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:
 * $\ds \bigcap \TT \in \SS$

Hence, from Set of Subsets which contains Set and Intersection of Subsets is Complete Lattice:


 * $\struct {\SS, \subseteq}$ is a complete lattice

where $\ds \bigcap \TT$ is the infimum of $\TT$.