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}$

Hoops

Polrims

Commutative residuated partially ordered monoids

BCK-algebras reduced type