Abbreviation: Pocrim
A \emph{pocrim} (short for \emph{partially ordered commutative residuated integral monoid}) is a structure $\mathbf{A}=\langle A,\oplus,\ominus,0\rangle$ of type $\langle 2,2,0\rangle$ such that
(1): $((x \ominus y) \ominus (x \ominus z)) \ominus (z \ominus y) = 0$
(2): $x \ominus 0 = x$
(3): $0 \ominus x = 0$
(4): $(x \ominus y) \ominus z = x \ominus (z \oplus y)$
(5): $x \ominus y = y \ominus x = 0 \Longrightarrow x=y$
Let $\mathbf{A}$ and $\mathbf{B}$ be pocrims. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \oplus y)=h(x) \oplus h(y)$, $h(x \ominus y)=h(x) \ominus h(y)$, $h(0)=0$.
A \emph{pocrim} is a structure $\mathbf{A}=\langle A,\oplus,\ominus,0\rangle$ such that
$\langle A,\ominus,0\rangle$ is a BCK-algebra
$(x \ominus y) \ominus z = x \ominus (z \oplus y)$
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.
1,1,2,7,26,129
$\begin{array}{lr}
f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &7\\ f(5)= &26\\
\end{array}$ $\begin{array}{lr}
f(6)= &129\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\
\end{array}$