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# 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 $(M,\circ, 1,\leq)$ that is left unital ($1\circ a = a$ for all $a\in M$), and has a binary operation, “implication”, written $b\to c$ satisfying $a \leq b\to c$ if and only if $a\circ b \leq c$.

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.