Special Semigroups Research Articles

The Endomorphism Semigroup of a Special Semigroup

The endomorphism semigroup for a class of commutative semigroups, called special semigroups, will be studied their structures will be determined in some important cases. AMS Classification : 20 Keywords: Special semigroups, freeness, divisibility, direct sums, endomorphism, endomorphism semigroup. doi: 10.3329/jbas.v32i1.2442 Journal of Bangladesh Academy of Sciences, Vol. 32, No. 1, 55-60, 2008

Propri�t�s de continuit� fine des fonctions coexcessives

We show at first that if (P t) is a special standard semi-group on a locally compact space E with a countable base and if u is an excessive function, then the semi-group $$\left( {\tilde P_t^{\left( u \right)} } \right)$$ , superharmonic transform of (P t) by u, is special standard if (and only if) the function u is regular. Then we prove that if two standard semi-groups in duality verifying Kunita-Watanabe's hypotheses are given and if g is a coexcessive function (excessive for the dual semi-group) then: 1) g is Borel measurable, 2) g is a. s. left continuous and has right limits on the sample path of the process $$\left( {X_{t^ - } } \right)_{0 < t < \zeta ,}$$ 3) g has a.s. right and left limits on the sample path of the process $$\left( {X_t } \right)_{0 < t < \zeta }$$ and 4) g is finely continuous except perhaps on a semi-polar set. If we suppose also that the second semi-group is special standard and that the coexcessive function g is regular, then 1) g is a.s. right continuous and has left limits on the sample paths of the process $$\left( {X_t } \right)_{0 < t < \zeta }$$ , and 2) g is finely continuous except perhaps on a polar set.