  An algebra $\mathbf A = \langle A, \to, 0 \rangle$, where $\to$ is binary and
$0$ is a constant, is called an implication zroupoid ($\mathcal I$-zroupoid,
for short) if $\mathbf A$ satisfies the identities: $(x \to y) \to z \approx
[(z' \to x) \to (y \to z)']'$ and $ 0'' \approx 0$, where $x' : = x \to 0$, and
$\mathcal I$ denotes the variety of all $\mathcal I$-zroupoids. An $\mathcal
I$-zroupoid is symmetric if it satisfies $x'' \approx x$ and $(x \to y')'
\approx (y \to x')'$. The variety of symmetric $\mathcal I$-zroupoids is
denoted by $\mathcal S$. An identity $p \approx q$, in the groupoid language
$\langle \to \rangle$, is called an identity of associative type of length $3$
if $p$ and $q$ have exactly 3 (distinct) variables, say x,y,z, and are grouped
according to one of the two ways of grouping: (1) $\star \to (\star \to \star)$
and (2) $(\star \to \star) \to \star$, where $\star$ is a place holder for a
variable. A subvariety of $\mathcal I$ is said to be of associative type of
length $3$, if it is defined, relative to $\mathcal I$, by a single identity of
associative type of length $3$. In this paper we give a complete analysis of
the mutual relationships of all subvarieties of $\mathcal I$ of associative
type of length $3$. We prove, in our main theorem, that there are exactly 8
such subvarieties of $\mathcal I$ that are distinct from each other and
describe explicitly the poset formed by them under inclusion. As an application
of the main theorem, we derive that there are three distinct subvarieties of
the variety $\mathcal S$, each defined, relative to $\mathcal S$, by a single
identity of associative type of length $3$.

Logic