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Sur quelques algèbres de la logique. (French) Zbl 0581.03046

A Hilbert (implicative) algebra is a set A with a binary operation \(\to\) on A satisfying: \((a\to a)\to a=a\); \(a\to a=b\to b\); \(a\to (b\to c)=(a\to b)\to (a\to c)\); (a\(\to b)\to ((b\to a)\to a)=(b\to a)\to ((a\to b)\to b)\). Put \(1=a\to a\). (These are the ”positive implicative algebras” in the wording of H. Rasiowa [An algebraic approach to non-classical logic (1974; Zbl 0299.02069)]. The main result regards the insertion of a ”conjunction” \(\wedge\) in a Hilbert algebra. So a Hertz algebra is a system (A,\(\to,\wedge)\) satisfying: \(a\to a=b\to b\); \((a\to b)\wedge b=b\); \(a\wedge (a\to b)=a\wedge b\); \(a\to (b\wedge c)=(a\to c)\wedge (a\to b)\). A Hertz algebra free on A, where A is a Hilbert algebra, is given as (A,f,HA) where HA is a Hertz algebra, f is a \(\to\)-embedding of A into HA, HA is generated by f(A) and if h is a \(\to\)-homomorphism from A into a Hertz algebra H, then there is a monomorphism h’ from HA into H such that \(h'\circ f=h\). Such a HA is unique up to isomorphism. To construct HA, first define \[ X\to Y=\cup^{m}_{i=1}\{x_ 1\to (...(x_ n\to y_ i)...)\} \] for \(X=\{x_ 1,...,x_ n\}\), \(Y=\{y_ 1,...,y_ m\}\) finite nonempty subsets of A, and \(I=\{1\}\). The relation \(''X\to Y=I\) and \(Y\to X=I''\) is an equivalence, and the quotient set naturally becomes a Hertz algebra (\(\wedge\) is defined through set theoretical union); that is HA. (This method is credited to Skolem for the implicative logical calculus).
Reviewer: A.Ursini

MSC:

03G25 Other algebras related to logic

Citations:

Zbl 0299.02069
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