nLab
Ackermann groupoid

Contents

Contents

Idea

An Ackermann groupoid is a particular type of algebraic mathematical structure that provides semantics for a flavour of relevance logic, a weak form of substructural logic.

Note that “groupoid” here does not mean groupoid, but magma. The terminology comes from logic, rather than category theory.

Definition

An Ackermann groupoid is a partially ordered magma (M,,1,)(M,\circ, 1,\leq) that is left unital (1a=a1\circ a = a for all aMa\in M), and has a binary operation, “implication”, written bcb\to c satisfying abca \leq b\to c if and only if abca\circ b \leq c.

This might be called an implicational Ackermann groupoid, since it provides semantic models for an implicational fragment of logic, together with intensional conjunction (here \to models implication, analogous to linear implication in linear logic). A positive Ackermann groupoid upgrades the underlying poset to a distributive lattice, permitting the interpretation of additional logical connectives, namely (classical) logical conjunction and logical disjunction.

Example

Every Church monoid is an Ackermann groupoid.

References

Ackermann groupoids were introduced in

  • Robert K. Meyer and Richard Routley, Algebraic analysis of entailment I, Logique et Analyse NOUVELLE SÉRIE, Vol. 15, No. 59/60 (1972) pp407-428, JSTOR

and named for Wilhelm Ackermann.

Last revised on April 30, 2021 at 03:04:24. See the history of this page for a list of all contributions to it.