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Unification in some substructural logics of BL-algebras and hoops. (English) Zbl 1156.03022
The paper is not sufficiently self-contained and organized to be easily read.
Unification in an equational theory \(E\) is the process of finding a substitution \(\sigma\) of individual variables that makes two given terms \(t_1, t_2\) equal, or unified, modulo the theory \(E\), i.e. \(\vdash_E \sigma t_1=\sigma t_2.\) Such a substitution is called a unifier for \(t_1\) and \(t_2\).
The unification type of a theory \(E\) can be unitary, finitary, infinitary or nullary.
The paper studies the unification type of the substructural logics associated to some partially ordered (\(k\)-potent, i.e. \(x^{k+1}=x^k\), for every \(x\), with \(k\geq 1\)) algebras containing the residuated pair \((\cdot, \Rightarrow)\) (i.e. \(x \cdot y \leq z \Longleftrightarrow x \leq y \Rightarrow z\), for all \(x,y,z\)) as a reduct: 4mm
hoops, basic hoops, Wajsberg hoops – which do not have a negation, because they do not have \(0\), the smallest element;
BL-algebras, Gödel algebras, Wajsberg algebras (MV-algebras) – which have a negation \(\neg x=x \Rightarrow 0\), because they have \(0\).
The results are: 4mm
the logics of \(k\)-potent hoops and the logics of \(k\)-potent BL-algebras (hence Gödel logic and the finite-valued Łukasiewicz logics) have unitary unification (with transparent unifiers); while
Basic Logic (the logic of BL-algebras) and the \(\infty\)-valued Łukasiewicz logic Ł\(_{\infty}\) (the logic of Wajsberg algebras (MV-algebras)) do not have unitary unification; consequently,
the logics of \(k\)-potent hoops are hereditarily structurally complete and the \(k\)-potent logics containing Basic Logic are structurally complete in the restricted sense; while
Basic Logic itself and Ł\(_{\infty}\) are not structurally complete, even in the restricted sense.

MSC:
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03G25 Other algebras related to logic
06D35 MV-algebras
06A12 Semilattices
06B99 Lattices
06F05 Ordered semigroups and monoids
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