dmonoid in Noam Zeilberger

#Contents#
* table of contents
{:toc}

## Idea

A _dmonoid_ is a "directed" monoid.

## Definition

A (left) **dmonoid** is a preorder $(X,\le)$ equipped with:

1. a binary operation $\cdot$ (called _multiplication_) which is monotonic in both arguments:
\[
  \label{mopsetbifunctor}
  \array{
  \arrayopts{\rowlines{solid}}
  a_1 \le a_2 \quad b_1 \le b_2 \\
  a_1\cdot b_1 \le a_2\cdot b_2
  }
\]
and which satisfies the **semi-associative law**:
\[
  \label{mopsetassoc}
  (a \cdot b)\cdot c \le a \cdot (b\cdot c)
\]
for all $a,b,c \in X$; and
2. an element $I \in X$ (called _unit_) satisfying left and right **unit laws**:
\[
  \label{mopsetunit}
  I \cdot a \le a
\]
\[
  \label{mopsetrunit}
  a \le a \cdot I
\]
for all $a \in X$.

## Related concepts

* [[imploid]]

## References

* {#Street2013} Ross Street. Skew-closed categories. _Journal of Pure and Applied Algebra_ 217(6) (June 2013), [arXiv:1205.6522](https://arxiv.org/abs/1205.6522)

* {#Szlachanyi2012} Kornel Szlachanyi. Skew-monoidal categories and bialgebroids. [arXiv:1201.4981](https://arxiv.org/abs/1201.4981)

* {#TamariPhD} Dov Tamari. _Monoïdes préordonnés et chaînes de Malcev_. Thèse, Université de Paris, 1951.

* {#Tamari1964} Dov Tamari. Sur quelques problèmes d'associativité. _Ann. sci. de Univ. de Clermont-Ferrand 2, Sér. Math._, vol. 24, pp. 91-107, 1964. [url](http://www.numdam.org/item/ASCFM_1964__24_3_91_0)

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