Semi-abelian by Design: Johnstone Algebras Unifying Implication and Division
David Forsman
TL;DR
Johnstone algebras provide a unified semi-abelian framework for combining implication-type and division-type operations in a single algebraic theory over $(*,\to,e)$. The paper defines a weak relative closure term $t(x,y)$ and shows how it yields equational control of order via $x\le y$, with an equational anti-symmetry criterion $v(x,y)\approx t(v(x,y),y)$, while proving that MC-algebras form a variety. It then develops the categorical-algebraic side, introducing Johnstone protomodular terms that ensure semi-abelianity, and explores enriching extensions by Residuation and Compositionality to recover residuated monoids and Galois connections, yielding RBJ and RBCJ algebras. No-go results show that balanced or inflationary/monotone theories cannot admit Malcev terms, clarifying the necessary structural constraints for semi-abelian behavior. The work situates Johnstone algebras as a bridge between groups, hoops, and Heyting-like logics within a robust equational/categorical framework, with potential for broader connections to residuated structures and dualities.
Abstract
Johnstone demonstrated that Heyting semilattices form a semi-abelian category via a specific triple of terms. Inspired by this work, we introduce \emph{Johnstone algebras} or J-algebras. The algebraic $(*,\to,e)$-theory $J$ of arities $(2,2,0)$ consists of three axioms carefully chosen to ensure protomodularity in alignment with Johnstone's terms. Johnstone algebras generalize well-known structures such as groups (division) and Heyting semilattices (implication) providing a unified framework within the well-behaved setting of semi-abelian categories. We present two primary contributions. First, we identify the M-axiom, \[ (t(x,y)\to x)\to (t(x,y)\to z) \approx x\to z, \text{ where }t(x,y) = (x\to y)\to y. \] The M-axiom is satisfied by residuated Johnstone algebras, and it can be considered a weakening of the H-axiom to comparable elements. We show that $t(x,y)$ defines a \emph{relative closure term} in MBC-algebras, and it implies that MBC-algebras form a variety of algebras, thereby generalizing the corresponding theorem related to HBCK-algebras. Second, we prove several no-go results, demonstrating that balanced theories or theories admitting non-discrete monotone or inflationary algebras cannot possess Malcev terms. Together, these results establish Johnstone algebras as significant structures that achieve desirable categorical properties by carefully integrating both logical and symmetric features, while closely avoiding the constraints imposed by our no-go results.