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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.

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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