# Is the class of Heyting algebras originating from directed graphs a variety?

The category RefGph of reflexive directed graphs is the functor category $\hat{∆}_1=\mbox{Fun}(∆^◦_1,$Set), where $∆_1$ is the simplex category truncated at level 1.

Hence the poset Sub(X) of subobjects of such a graph X is a Heyting algebra.

I am interested in the category $K_1$, whose objects are the Heyting algebras, which are isomorphic to those of the form Sub(X) for some reflexive directed graph $X$.

As F. Dorais explains, these algebras can be described as subalgebras of powers of the Heyting algebra $\Omega_1$, whose underlyinig set consists of the five elements $$|\Omega_1|=\{0, s,t,s\vee t,1\}.$$ Its lattice operations are given by the tables $$\begin{array}{c|ccccc} \wedge & 0 & s & t & s\vee t & 1 \\\hline 0 & 0 & 0 & 0 & 0 & 0 \\ s & 0 & s & 0 & s & s \\ t & 0 & 0 & t & t & t \\ s\vee t & 0 & s & t & s\vee t & s\vee t \\ 1 & 0 & s & t & s\vee t & 1 \\ \end{array}\qquad\qquad \begin{array}{c|ccccc} \vee & 0 & s & t & s\vee t & 1 \\\hline 0 & 0 & s & t & s\vee t & 1 \\ s & s & s & s\vee t & s\vee t & 1 \\ t & t & s\vee t & t & s\vee t & 1 \\ s\vee t & s\vee t & s\vee t & s\vee t & s\vee t & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array}$$

So its underlying poset has the Hasse-diagram

Its (relative) pseudocomplement is given by the table

$$\begin{array}{c|ccccc} \Rightarrow & 0 & s & t & s\vee t & 1 \\\hline 0 & 1 & 1 & 1 & 1 & 1 \\ s & t & 1 & t & 1 & 1 \\ t & s & s & 1 & 1 & 1 \\ s\vee t & 0 & s & t & 1 & 1 \\ 1 & 0 & s & t & s\vee t & 1 \\ \end{array}\qquad\qquad \begin{array}{c|c} \neg & \\\hline 0 & 1 \\ s & t \\ t & s \\ s\vee t & 0 \\ 1 & 0 \\ \end{array}$$

Hence my question:

Is $K_1$ a variety of algebras? (finitary algebraic category)

Can you give some references to relevant papers?

• As I mentioned in my earlier answer, since every $\operatorname{Sub}(X)$ is complete, $K_1$ isn't closed under subalgebras so it is not a variety. – François G. Dorais Mar 11 '16 at 19:55