Some Theorems on Universal Algebras. II

Fuzzy universal algebras on $L$-sets

This paper attempts to generalize universal algebras on classical sets to $L$-sets when $L$ is a GL-quantale. Some basic notions of fuzzy universal algebra on an $L$-set are introduced, such as subalgebra, quotient algebra, homomorphism, congruence, and direct product etc. The properties of them are studied. $L$-valued power algebra is also introduced and it is shown there is an onto homomorphi...

Some Characterization Theorems for Infinitary Universal Horn Logic Without Equality

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Some Theorems on Coverings

Some Theorems on Continua

PROOF. Let {Gn} and {Hn} denote monotonie decreasing sequences of open sets which close down upon .Pi and F2 respectively. We may suppose that d and Hi are so chosen that G i # i = 0. Now, by Mullikin's theorem, there is a connected subset of C— C(Gw+27n) which has a limit point in Gn and a limit point in Hn. The closure of such a connected set is a subcontinuum of C which "extends" from Gn to ...

Some Theorems on Subseries

1. Absolutely convergent series. A simple calculation reveals that the arithmetic mean value of all subsums (including the void sum) of a given finite sum sn=ui+U2 + • • • +un is equal to sn/2. In this section we shall show (see Theorem 1 below) that an integral mean value can be found, consistent with the preceding, for the sums of all infinite subseries of a given absolutely convergent series...