The half-open real unit interval an automorphism-invariant positive normalized linear functional onH. SinceHis 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, 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.

In this paper, we mainly investigate the existence of states based on the Glivenko theorem in bounded semihoops, which are building blocks for the algebraic semantics for relevant fuzzy logics. First, we extend algebraic formulations of the Glivenko theorem to bounded semihoops and give some characterizations of Glivenko semihoops and regular semihoops. The category of regular semihoops is a reflective subcategory of the category of Glivenko semihoops. Moreover, by means of the negative translation term, we characterize the Glivenko variety. Then (...) we show that the regular semihoop of regular elements of a free algebra in the variety of Glivenko semihoops is free in the corresponding variety of regular semihoops. Similar results are derived for the semihoop of dense elements of free Glivenko semihoops. Finally, we give a purely algebraic method to check the existence of states on Glivenko semihoops. In particular, we prove that a bounded semihoop has Bosbach states if and only if it has a divisible filter, and a bounded semihoop has Riečan states if and only if it has a semi-divisible filter. (shrink)

We present a stronger variation of state MV-algebras, recently presented by T. Flaminio and F. Montagna, which we call state-morphism MV-algebras. Such structures are MV-algebras with an internal notion, a state-morphism operator. We describe the categorical equivalences of such state MV-algebras with the category of unital Abelian ℓ-groups with a fixed state operator and present their basic properties. In addition, in contrast to state MV-algebras, we are able to describe all subdirectly irreducible state-morphism MV-algebras.