Measures induced by units

The half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop -equivalently, in the enveloping lattice-ordered abelian group- amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoop H induces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional on H. Since H is representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals -in this context usually called states- amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group of H), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.Comment: 24 pages, 1 figure. Revised version according to the referee's suggestions. Examples added, proof of Lemma 2.6 simplified, Section 7 expanded. To appear in the Journal of Symbolic Logi

Coherent algebras and noncommutative projective lines

A well-known conjecture says that every one-relator group is coherent. We state and partly prove an analogous statement for graded associative algebras. In particular, we show that every Gorenstein algebra $A$ of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line \PP^1 as a noncommutative scheme based on the coherent noncommutative spectrum \cohp A of such an algebra $A$, that is, the category of coherent $A$-modules modulo the torsion ones. This category is always abelian Ext-finite hereditary with Serre duality, like the category of coherent sheaves on \PP^1. In this way, we obtain a sequence \PP^1_n ($n\ge 2$) of pairwise non-isomorphic noncommutative schemes which generalize the scheme \PP^1 = \PP^1_2.Comment: 10 pages. In this version, Prop. 1.5 extended, few comments added et

Generic substitutions

Up to equivalence, a substitution in propositional logic is an endomorphism of its free algebra. On the dual space, this results in a continuous function, and whenever the space carries a natural measure one may ask about the stochastic properties of the action. In classical logic there is a strong dichotomy: while over finitely many propositional variables everything is trivial, the study of the continuous transformations of the Cantor space is the subject of an extensive literature, and is far from being a completed task. In many-valued logic this dichotomy disappears: already in the finite-variable case many interesting phenomena occur, and the present paper aims at displaying some of these.Comment: 22 pages, 2 figures. Revised version according to the referee's suggestions. To appear in the J. of Symbolic Logi