Noam Zeilberger dmonoid (Rev #1)

Contents

Idea

A dmonoid is a “directed” monoid.

Definition

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

  1. a binary operation \cdot (called multiplication) which is monotonic in both arguments:
    (1)a 1a 2b 1b 2 a 1b 1a 2b 2 \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)(ab)ca(bc) (a \cdot b)\cdot c \le a \cdot (b\cdot c)

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

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

    for all aXa \in X.

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

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