M. Sion and T. Traynor investigated ([15]-[17]), measures and integrals having values in topological groups or semigroups. Their definition of integrability was a modification of Phillips-Rickart bilinear vector integrals, in locally convex topological vector spaces.The purpose of this paper is to develop a good notion of an integration process in partially ordered groups, based on their order structure. The results obtained generalize some of the results of J. D. M. Wright ([19]-[22]) where the measurable functions are real-valued and the measures take values in partially ordered vector spaces.Let if be a σ-algebra of subsets of T, X a lattice group, Y, Z partially ordered groups and m : H → F a F-valued measure on H. By F(T, X), M(T, X), E(T, X) and S(T, X) are denoted the lattice group of functions with domain T and with range X, the lattice group of (H, m)-measurable functions of F(T, X) and the lattice group of (H, m)-elementary measurable functions of F(T, X) and the lattice group of (H, m)-simple measurable functions of F(T, X) respectively.

a Communicated by I. StratisWe study functional differential evolution equations of the formwhere A is infinitesimal generator of an analytic semigroup in a Banach space E (with or without order) and the given right-hand side modelling delay. In many cases, E is a Banach space of sections, such as vector field or differential forms of a (real or complex) vector bundle( of possibly infinite dimension) over a locally convex manifold, for example, a Carathéodory-Finsler manifold. The operator A may in particular be generated by a Dirichlet form acting on an ordered Hilbert space. As an application, we consider a problem from thermo-magnetohydrodynamics.

Denning a Radon-type integration process we extend the Alexandras', Fichtengolts-KantorovichHildebrandt and Riesz integral representation theorems in partially ordered vector spaces.We also identify some classes of operators with other classes of operator-valued set functions, the correspondence between operator and operator-valued set function being given by integration.All these established results can be immediately applied in C*-algebras (especially in W*-algebras and AW '-algebras of type I), in Jordan algebras, in partially ordered involutory (0*-)algebras, in semifields, in quantum probability theory, as well as in the operator FeynmanKac formula.

Abstract.The present paper is concerned with partially ordered semigroup-valued measures. 1. Preliminaries. By a partially ordered semigroup X we mean a commutative semigroup with identity 0, equipped by a partial ordering <, compatible with the structure of X under the conditions:(i) If x, y, z are elements of X with x < y (x < y and x ^ y) then x + z < y + z.(ii) x + sup E = sup (x + E), whenever there exist sup E (the supremum of E in X) and sup(x + E),E E X,x E X. Now X is monotone complete if every majorised increasing directed family in X has a supremum in X. Moreover, X is of the countable type if every subset £ of A" that has a supremum in X, contains a countable subset E* Ç E so that: sup E = sup E*.Let A" be a partially ordered semigroup and H a ring of subsets of T. The function m: H -» X is an o-measure (order measure) on H, if m is positive on H imiA) > 0, for every A in H) and mi\J"eNAn) = sup{2^_,w(^,): « E N} whenever iAn)neN is a disjoint sequence of elements of H with ( U "eAi^,) E H.The following propositions can be easily proved. Proposition 1.1. Let m: H -^ X be an o-measure on H.(1)«j(0) = 0.(2) m is finitely additive on H and miA) < miB), whenever A, B E H with A E B.(3) For every sequence iA")neN in H with (U"ejv^n) e H and sup{2ï_xmiAi): n E N) E X, implies: mi(JneNAn) < sup{2?=1m(v4,.): « 6 N).Received by the editors August 22, 1977 and, in revised form, November 30, 1977 AMS iMOS) subject classifications (1970). Primary 46G99; Secondary 28A55.Key words and phrases. Partially ordered semigroup, monotone complete partially ordered semigroup, partially ordered semigroup of the countable type, o-measure, absolutely continuous and singular o-measure, partially ordered topological semigroup, o-compatible topology with the partial ordering, T^-measure.

Abstract.The present paper is concerned with partially ordered semigroup-valued measures. 1. Preliminaries. By a partially ordered semigroup X we mean a commutative semigroup with identity 0, equipped by a partial ordering <, compatible with the structure of X under the conditions:(i) If x, y, z are elements of X with x < y (x < y and x ^ y) then x + z < y + z.(ii) x + sup E = sup (x + E), whenever there exist sup E (the supremum of E in X) and sup(x + E),E E X,x E X. Now X is monotone complete if every majorised increasing directed family in X has a supremum in X. Moreover, X is of the countable type if every subset £ of A" that has a supremum in X, contains a countable subset E* Ç E so that: sup E = sup E*.Let A" be a partially ordered semigroup and H a ring of subsets of T. The function m: H -» X is an o-measure (order measure) on H, if m is positive on H imiA) > 0, for every A in H) and mi\J"eNAn) = sup{2^_,w(^,): « E N} whenever iAn)neN is a disjoint sequence of elements of H with ( U "eAi^,) E H.The following propositions can be easily proved. Proposition 1.1. Let m: H -^ X be an o-measure on H.(1)«j(0) = 0.(2) m is finitely additive on H and miA) < miB), whenever A, B E H with A E B.(3) For every sequence iA")neN in H with (U"ejv^n) e H and sup{2ï_xmiAi): n E N) E X, implies: mi(JneNAn) < sup{2?=1m(v4,.): « 6 N).Received by the editors August 22, 1977 and, in revised form, November 30, 1977 AMS iMOS) subject classifications (1970). Primary 46G99; Secondary 28A55.Key words and phrases. Partially ordered semigroup, monotone complete partially ordered semigroup, partially ordered semigroup of the countable type, o-measure, absolutely continuous and singular o-measure, partially ordered topological semigroup, o-compatible topology with the partial ordering, T^-measure.