Typed Congruences and Quotient-Critical Irreducibility in Complete Ternary $\Gamma$-Semirings

Abstract

We study congruences and binary reducibility in completely additive ternary $\Gamma$-semirings through their two-sorted atomic cores. For an odd $m\ge 3$, an abelian group $G$ of exponent dividing $m-1$, and $u,v\in G$, we construct a two-branch symmetric $m$-ary band $F_m(G;u,v)$. We prove \[ \Con(F_m(G;u,v))\cong\operatorname{Sub}(G)\times B_2 \] and classify every carrier--index congruence pair: $(\theta ,\varphi )$ is typed exactly when $\theta$ is an ordinary congruence and $\varphi$ refines $\theta$. We also give a quotient-by-quotient reducibility criterion. In the finite case, $F_m(G;u,v)$ is quotient-critically irreducible exactly when $G$ is a cyclic $p$-group and $u-v$ has order $p$. The specialization $G={\mathbb{Z}}_2$ yields a four-point family $H_m$ extending the irreducible ternary example of Devillet and Mathonet. Its congruence lattice is $B_3$, its alternating core has exactly $36$ typed congruence pairs, and its automorphism group is $C_2$. Its seven proper quotients are reducible and form five isomorphism types; in arity five their exact reduction counts are determined. The powerset lift of $H_5$ is a $16$-element atom-total complete atomic Boolean ternary $\Gamma$-semiring with no atomic binary collapse, whereas each proper diagonal strong atom-saturated quotient has one. Its parity congruence is recovered on atoms by one explicitly specified depth-one polynomial inequation.