### A Hölder-type inequality for positive functionals on $\text{\Phi}$-algebras.

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Non-Archimedean f-rings need not be p-distributive. Moreover, if {di|i} is a subset of a non-Archimedean f-ring and a ≥ 0, the elements a vi di and vi adi need not be equal. We prove, however, that the difference is an infinitely small element when the ring has a strong unity.

A DC-space (or space of dense constancies) is a Tychonoff space $X$ such that for each $f\in C\left(X\right)$ there is a family of open sets $\{{U}_{i}\phantom{\rule{0.222222em}{0ex}}i\in I\}$, the union of which is dense in $X$, such that $f$, restricted to each ${U}_{i}$, is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean $f$-algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions...

Using lattice-ordered algebras it is shown that a totally ordered field which has a unique total order and is dense in its real closure has the property that each of its positive semidefinite rational functions is a sum of squares.

This work discusses the problem of Arens regularity of a lattice-ordered ring. In this prospect, a counterexample is furnished to show that without extra conditions, a lattice-ordered ring need not be Arens regular. However, as shown in this paper, it turns out that any $f$-ring in the sense of Birkhoff and Pierce is Arens regular. This result is then used and extended to the more general setting of almost $f$-rings introduced again by Birkhoff.

We show some families of lattice effect algebras (a common generalization of orthomodular lattices and MV-effect algebras) each element E of which has atomic center C(E) or the subset S(E) of all sharp elements, resp. the center of compatibility B(E) or every block M of E. The atomicity of E or its sub-lattice effect algebras C(E), S(E), B(E) and blocks M of E is very useful equipment for the investigations of its algebraic and topological properties, the existence or smearing of states on E, questions...

In this paper we give necessary and sufficient conditions in order that a contractive projection on a complex $f$-algebra satisfies Seever’s identity.