Abbreviation: Frm

A \emph{frame} is a structure $\mathbf{A}=\langle A, \bigvee, \wedge, e, 0\rangle$ of type $\langle\infty, 2, 0, 0\rangle$ such that

$\langle A, \bigvee, 0\rangle$ is a complete semilattice with $0=\bigvee\emptyset$,

$\langle A, \wedge, e\rangle$ is a meet semilattice with identity, and

$\wedge$ distributes over $\bigvee$: $x\wedge(\bigvee Y)=\bigvee_{y\in Y}(x\wedge y)$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

Let $\mathbf{A}$ and $\mathbf{B}$ be frames. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(\bigvee X)=\bigvee h[X]$ for all $X\subseteq A$ (hence $h(0)=0$), $h(x \wedge y)=h(x) \wedge h(y)$ and $h(e)=e$.

A \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Example 1:

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

$\begin{array}{lr}

f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\

\end{array}$ $\begin{array}{lr}

f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

[[...]] subvariety

[[...]] expansion

[[...]] supervariety

[[...]] subreduct