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A relation $R$ on a set $A$ is reflexive iff for every $a\in A$ we have $(a,a)\in R$. Note that this means that unlike transitivity and symmetry, reflexivity crucially depends on the "carrier set" involved". So for example, $\{(1,1),(2,2),(1,2)\}$ is reflexive as a relation on $\{1,2\}$ but is not reflexive as a relation on $\{1,2,3\}$. Note ...


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Some people apparently call a homomorphism (not necessarily surjective) satisfying this condition a "strong homomorphism". See this question and my answer there. As explained in the linked question, the terminology "strong homomorphism" can also refer to a homomorphism which reflects relations: for all $\overline{a}\in A$, $R^A(\overline{...


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Suppose that $\mathcal{H}$ is any lower bound of $\{\mathcal{F},\mathcal{G}\}$. I'll denote the refinement partial order by $\le$. Let $H \in \mathcal{H}$. It is the case that $\mathcal{H} \le \mathcal{F}$ so there exists some $F_1 \in \mathcal{F}$ such that $H \subseteq F_1$. Also $\mathcal{H} \le \mathcal{G}$ so there is a $G_1 \in \mathcal{G}$ such that $...


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