Let F denote a totally real number field of degree d > 1 over Q, let p be a prime number and let χ be a totally odd Hecke character of finite order of F . Klingen and Siegel have shown that the values of the Hecke L-series L(χ, s) at integers n ≤ 0 lie in the algebraic closure Q ⊆ C of Q. In [14] Shintani gave another proof by constructing a nice fundamental domain (i.e. a finite disjoint union...

The framework. In 1956, G. Birkhoff G. and R.S. Pierce [1] conjectured the following: do the algebra of piecewise polynomial functions with real coefficients and the algebra of functions that can be written as Inf and Sup of finitely many polynomials coincide? The conjecture – that can be equivalently stated asking for the free n-generated lattice-ordered algebra to be isomorphic with the algeb...

Theorem 2 is also consequence of Theorem 1, so the two theorems are equivalent. In fact, the implication Theorem 1 ⇒ Theorem 2 is usually how one first meets the fundamental theorem of algebra in a linear algebra course: it assures us that any complex square matrix has an eigenvector because the characteristic polynomial of the matrix has a complex root. But here, we will prove Theorem 2 withou...

In this study, we define the generalized symmetric bi-derivation and its trace on MV-algebras give examples. We investigate some properties of bi-derivation’s trace. introduce isotone bi-additive bi-derivation. show that if A is a linearly ordered MV-algebra, Γ related to D γ , then = 0 or γ1 1

Georgescu and Iorgulescu [3] introduced pseudo-BCK-algebras (in a slightly different way) as a non-commutative generalization of BCK-algebras, in the sense that if →= , then the algebra (A,→, 1) is a BCK-algebra. Pseudo-BCK-algebras relate to (non-commutative) residuated lattices as BCK-algebras do to commutative residuated lattices; specifically, by [6], pseudoBCK-algebras are just the 〈→, , 1...

We study observables as r-D-homomorphisms de®ned on Boolean D-posets of subsets into a Boolean Dposet. We show that given an atomic r-complete Boolean D-poset P with the countable set of atoms there exist a r-complete Boolean D-poset of subsets S and a r-D-homomorphism h from S onto P, more precisely we can choose S ˆ B…R†, which gives an analogy of the Loomis± Sikorski representation theorem f...