The Downward Transfer of Elementary Satisfiability of Partition Logics

The following operation has frequently occurred in mathematics and its applications: partition a set into several disjoint non-empty subsets, so that the elements in each partition subset are homogeneous or indistinguishable with respect to some given properties. H.-D.Ebbinghaus first (in 1991) distilled from this phenomenon, which is a monadic second order property in nature, a special kind of quantifiers, and augmented first order logic Lωω with them to obtain a family of extended logics, called monadic partition logics. Interesting applications have been found outside mathematics, especially in computer science [8, 10]. Ebbinghaus quantifiers (i.e. monadic partition quantifiers) will reduce to Malitz quantifiers [6] if we restrict them with some infinite cardinality requirements. However the latter one appeared earlier and their backgrounds are also different. There are several types of partition quantifiers, such as 2-partition or multi-partition, monadic or non-monadic type. When augmenting Lωω with all the monadic partition quantifiers, we get the extended logic L(MP); while LP denotes the extended logic obtained by adding all sorts of partition quantifiers to Lωω [8, 9, 10]. First we introduce the semantic interpretation of monadic partition quantifiers. As a typical example we look at a special case of type, P . Definition 1. A |= P 2,1 x,x′;yφ(x, x , y) iff there is a partition of the universe A(|A| ≥ 2) of A: A = A∪̇A, A 6= ∅ 6= A , such that for all a, a ∈ A, b ∈ A, A |= φ[a, a, b]. Obviously L(MP) can be embedded into MSO (monadic second order logic). The partition quantifiers of non-monadic type concern the partition of the Cartesian product of the universe, i.e. the partition of multi-dimensional space. For instance A |= P 1,1 2 φ(x, y, x , y) iff there is a partition of A: A = U0∪̇U1, U0 6= ∅ 6 = U1, such that for all (a0, b0) ∈ U0, (a1, b1) ∈ U1, A |= φ[a0, b0, a1, b1]. Surely it can be translated into SO (second order logic), so LP ≤ SO.