# Tag Info

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 ...
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 $\overline{a}\in A$, $R^A(\overline{... 1 Suppose that$\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}$such that$...