On complete $MV$-algebras
Ján Jakubík · Czechoslovak Mathematical Journal · 1995
Though the number of published papers on MV-algebras is rather large (the fundamental source are Chang's articles [1] and [2]), the terminology and notation in this field seem to be far from being unified.We will apply the terminology from [5], [6].It is well-known that MV-algebras are term equivalent to Wajsberg algebras (called also VV-algebras); cf., e.g., Cignoli [3].Further, MV-algebras are categorically equivalent to bounded commutative HCIf-algebras (cf.Mundici [8]); such HCI^-algebras were studied by Traczyk [10].Cignoli [3] studied the structure of MV-algebras which are complete and atomic.His main result is the following theorem:(*) ([3], Theorem 2.6.)An MV-algebra is complete and atomic if and only if it is a direct product of finite linearly ordered MV-algebras.An MV-algebra srf which is a direct product of MV-algebras srfi (i G I) is complete if and only if all st/i are complete.Further, a complete linearly ordered MV-algebra is atomic if and only if it is finite (cf.1.3 below).Thus (*) can be expressed as follows:(**) An MV-algebra is complete and atomic if and only if it is a direct product of complete atomic linearly ordered algebras.Let srf -(A; 0, *, -i,0,1) be an MV-algebra.We can introduce lattice operations V, A, and hence also the corresponding partial order ^ on A (cf. Section 1 below).Let 0 1 be a cardinal.The element x will be called an a-atom of s/ if the interval [0,:r] is a chain having cardinality a. Hence the notion of the 2-atom coincides with the usual notion of the atom.The MV-algebra £/ is said to be a-atomic if for each 0 < y E A there exists an a-atom x of s/ with x ^ y.