MV -Algebras of Continuous Functions and l-Monoids

Abstract

Abstract. A. Di Nola & S.Sessa [8] showed that two compact spaces X and Y arehomeomorphic iff the MV -algebras C(X,I) and C(Y,I) of continuous functions defined onX and Y respectively are isomorphic. And they proved that A is a semisimple MV -algebraiff A is a subalgebra of C(X) for some compact Hausdorff space X. In this paper, firstlyby use of functorial argument, we show these characterization theorems. Furthermorewe obtain some other functorial results between topological spaces and MV -algebras.Secondly as a classical problem, we find a necessary and sufficient condition on a givenresiduated l-monoid that it is segmenently embedded into an l-group with order unit. 1. IntroductionAn MV-algebra is a universal algebra (A,+,·,∗,0,1) of (2,2,1,0,0) type suchthat (A,+,0) is an abelian monoid and moreover, x+1 = 1,x ∗∗ = x,0 ∗ = 1,x+x ∗ =1,x·y = (x ∗ +y ∗ ) and x+x ∗ y = y+y ∗ x for all x,y ∈ A. By setting x∨y = x+x ∗ yand x∧y = x(x ∗ +y) we have (A,∨,∧,0,1) as a bounded distributive lattice.The system of MV-algebras is a kind of better system in the sense that closedunder subalgebras, quotients and products and the free MV-algebra with a denu-merable set of generators can be described by MV-algebras of continuous I = [0,1]-valued functions on the Hilbert cube [11]. Furthermore, the variety of MV-algebrasis a Malcev variety and has the congruence regularity [10].In the first part of this paper, we establish a dual-adjunction (η,e) : S ‘ C fromthe subcategory X of Tychonoff spaces of Top into the subcategory A of semi-simple MV-algebras of M