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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:
   (1)$$\array{ \arrayopts{\rowlines{solid}} a_1 \le a_2 \quad b_1 \le b_2 \\ a_1\cdot b_1 \le a_2\cdot b_2 }$$

   and which satisfies the semi-associative law:

   (2)$$(a \cdot b)\cdot c \le a \cdot (b\cdot c)$$

   for all $a,b,c \in X$; and

2. an element $I \in X$ (called unit) satisfying left and right unit laws:
   (3)$$I \cdot a \le a$$
   (4)$$a \le a \cdot I$$

   for all $a \in X$.

Related concepts

• imploid

References

• Ross Street. Skew-closed categories. Journal of Pure and Applied Algebra 217(6) (June 2013), arXiv:1205.6522

• Kornel Szlachanyi. Skew-monoidal categories and bialgebroids. arXiv:1201.4981

• Dov Tamari. Monoïdes préordonnés et chaînes de Malcev. Thèse, Université de Paris, 1951.

• Dov Tamari. Sur quelques problèmes d’associativité. Ann. sci. de Univ. de Clermont-Ferrand 2, Sér. Math., vol. 24, pp. 91-107, 1964. url