nLab Ackermann groupoid

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.

Related concepts

• relevance monoidal category
• relevance logic
• Church monoid
• Heyting algebra

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.