## Sur quelques algĂ¨bres 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

Zbl 0299.02069
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