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HEREDITARILY STRUCTURALLY COMPLETE EXTENSIONS OF
${\mathbf {RM}}$
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 06 March 2025, pp. 1-22
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- Article
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This paper focuses on the structurally complete extensions of the system
$\mathbf {R}$-mingle (
$\mathbf {RM}$). The main theorem demonstrates that the set of all hereditarily structurally complete extensions of 
$\mathbf {RM}$ is countably infinite and forms an almost-chain, with only one branching element. As a corollary, we show that the set of structurally complete extensions of 
$\mathbf {RM}$ that are not hereditary is also countably infinite and forms a chain. Using algebraic methods, we provide a complete description of both sets. Furthermore, we offer a characterization of passive structural completeness among the extensions of 
$\mathbf {RM}$: specifically, we prove that a quasivariety of Sugihara algebras is passively structurally complete if and only if it excludes two specific algebras. As a corollary, we give an additional characterization of quasivarieties of Sugihara algebras that are passively structurally complete but not structurally complete. We close the paper with a characterization of actively structurally complete quasivarieties of Sugihara algebras.
