Mathematical Structures: Pocrims

# Pocrims

http://mathcs.chapman.edu/structuresold/files/Pocrims.pdf
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\begin{document}
\textbf{\Large Pocrims}

\abbreviation{Pocrim}

\begin{definition}
A \emph{pocrim} (short for \emph{partially ordered commutative residuated integral monoid}) is a structure $\mathbf{A}=\langle A,\oplus,\dotminus,0\rangle$ of type $\langle 2,2,0\rangle$ such that

(1):  $((x \dotminus y) \dotminus (x \dotminus z)) \dotminus (z \dotminus y) = 0$

(2):  $x \dotminus 0 = x$

(3):  $0 \dotminus x = 0$

(4):  $(x \dotminus y) \dotminus z = x \dotminus (z \oplus y)$

(5):  $x \dotminus y = y \dotminus x = 0 \implies x=y$

Remark: This is a template.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{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:
$h(x \oplus y)=h(x) \oplus h(y)$,
$h(x \dotminus y)=h(x) \dotminus h(y)$,
$h(0)=0$.
\end{morphisms}

\begin{definition}
A \emph{pocrim} is a structure $\mathbf{A}=\langle A,\oplus,\dotminus,0\rangle$ such that

$\langle A,\dotminus,0\rangle$ is a \href{BCK-algebras.pdf}{BCK-algebra}

$(x \dotminus y) \dotminus z = x \dotminus (z \oplus y)$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

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.

\begin{tabular}{|ll|}\hline
Classtype                       & quasivariety \cite{Higgs1984} \\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & \\\hline
Congruence modular              & \\\hline
Congruence $n$-permutable       & \\\hline
Congruence regular              & \\\hline
Congruence uniform              & \\\hline
Congruence extension property   & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong.     & \\\hline
Amalgamation property           & \\\hline
Strong amalgamation property    & \\\hline
Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} $f(n)=$ number of members of size $n$.

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

\end{finite_members}

\begin{subclasses}\

\href{Hoops.pdf}{Hoops}

\end{subclasses}

\begin{superclasses}\

\href{Polrims.pdf}{Polrims}

\href{Commutative_partially_ordered_residuated_monoids.pdf}{Commutative partially ordered residuated monoids}

\href{BCK-algebras.pdf}{BCK-algebras} reduced type

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Higgs1984}
D. Higgs, \emph{Dually residuated commutative monoids with identity element as least element do not form an equational class}, Math. Japon.,
\textbf{29}, 1984, no. 1, 69--75 \href{http://www.ams.org/mathscinet-getitem?mr=0737536}{MRreview}

\end{thebibliography}

\end{document}
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