Pocrims

Abbreviation: Pocrim

Definition

A pocrim (short for 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): $x \ominus y = y \ominus x = 0 \Longrightarrow x=y$

Morphisms

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: $h(x \oplus y)=h(x) \oplus h(y)$, $h(x \ominus y)=h(x) \ominus h(y)$, $h(0)=0$.

Definition

A 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:

Properties

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.

Classtype quasivariety 1)

Finite members

1,1,2,7,26,129

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &7\\ f(5)= &26\\ \end{array}$ $\begin{array}{lr} f(6)= &129\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

References

1) D. Higgs, Dually residuated commutative monoids with identity element as least element do not form an equational class, Math. Japon., 29, 1984, no. 1, 69–75 MRreview