Mathematics

All known structural extensions of the substructural logic F L e , Full Lambek calculus with exchange/commutativity, (corresponding to subvarieties of commutative residuated lattices axiomatized by {∨,⋅,1} -equations) have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.

Presburger Arithmetic is the true theory of natural numbers with addition. We study interpretations of Presburger Arithmetic in itself. The main result of this paper is that all self-interpretations are definably isomorphic to the trivial one. Here we consider interpretations that might be multi-dimensional. We note that this resolves a conjecture by A. Visser. In order to prove the result we show that all linear orderings that are interpretable in (N;+) are scattered orderings with the finite Hausdorff rank and that the ranks are bounded in the terms of the dimensions of the respective interpretations.

In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class ( R -mec), a special kind of multidimensional asymptotic class ( R -mac) with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatisation [14] to prove the following result (Theorem 4.6.4), as conjectured by Macpherson: For any countable language L and any positive integer d the class C(L,d) of all finite L -structures with at most d 4-types is a polynomial exact class in L , where a polynomial exact class is a multidimensional exact class with polynomial measuring functions.

This paper aims to build a new understanding of the nonstandard mathematical analysis. The main contribution of this paper is the construction of a new set of numbers, R Z < , which includes infinities and infinitesimals. The construction of this new set is done naïvely in the sense that it does not require any heavy mathematical machinery, and so it will be much less problematic in a long term. Despite its naïvety character, the set R Z < is still a robust and rewarding set to work in. We further develop some analysis and topological properties of it, where not only we recover most of the basic theories that we have classically, but we also introduce some new enthralling notions in them. The computability issue of this set is also explored. The works presented here can be seen as a contribution to bridge constructive analysis and nonstandard analysis, which has been extensively (and intensively) discussed in the past few years.

The Boolean logic of subsets, usually presented as `propositional logic,' is considered as being "classical" while intuitionistic logic and the many sublogics and off-shoots are "non-classical." But there is another mathematical logic, the logic of partitions, that is at the same mathmatical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense. Our purpose here is to explore the notions of negation and implication in that other mathematical logic of partitions.

It is well-known fact that there exists 1-1 correspondence between so-called double (or flou) sets and intuitionistic sets (also known as orthopairs). At first glance, these two concepts seem to be irreconcilable. However, one must remember that algebraic operations in these two classes are also defined differently. Hence, the expected compatibility is possible. Contrary to this approach, we combine standard definition of double set with operations which are typical for intuitionistic sets. We show certain advantages and limitations of this viewpoint. Moreover, we suggest an interpretation of our sets and operations in terms of logic, data clustering and multi-criteria decision making. As a result, we obtain a structure of discussion between several participants who propose their "necessary" and "allowable" requirements or propositions.

Robin Hirsch posed in 1996 the Really Big Complexity Problem: classify the computational complexity of the network satisfaction problem for all finite relation algebras A . We provide a complete classification for the case that A is symmetric and has a flexible atom; the problem is in this case NP-complete or in P. If a finite integral relation algebra has a flexible atom, then it has a normal representation B . We can then study the computational complexity of the network satisfaction problem of A using the universal-algebraic approach, via an analysis of the polymorphisms of B . We also use a Ramsey-type result of Nešetřil and Rödl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.

We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group Γ we introduce the Γ -jump. We study the elementary properties of the Γ -jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups Γ , the Γ -jump is \emph{proper} in the sense that for any Borel equivalence relation E the Γ -jump of E is strictly higher than E in the Borel reducibility hierarchy. On the other hand there are examples of groups Γ for which the Γ -jump is not proper. To establish properness, we produce an analysis of Borel equivalence relations induced by continuous actions of the automorphism group of what we denote the infinite Γ -tree, and relate these to iterates of the Γ -jump. We also produce several new examples of equivalence relations that arise from applying the Γ -jump to classically studied equivalence relations and derive generic ergodicity results related to these. We apply our results to show that the complexity of the isomorphism problem for countable scattered linear orders properly increases with the rank.

The present article studies nilpotent and Hamiltonian cancellative residuated lattices and their relationship with nilpotent and Hamiltonian lattice-ordered groups. In particular, results about lattice-ordered groups are extended to the domain of residuated lattices. The two key ingredients that underlie the considerations of this paper are the categorical equivalence between Ore residuated lattices and lattice-ordered groups endowed with a suitable modal operator; and Malcev's description of nilpotent groups of a given nilpotency class c in terms of a semigroup equation.

Let K be an elementary extension of Q p , V be the set of finite a∈K , st be the standard part map K m → Q m p , and X⊆ K m be K -definable. Delon has shown that Q m p ∩X is Q p -definable. Yao has shown that dim Q m p ∩X≤dimX and dimst( V n ∩X)≤dimX . We give new NIP -theoretic proofs of these results and show that both inequalities hold in much more general settings. We also prove the analogous results for the expansion Q an p of Q p by all analytic functions Z n p → Q p . As an application we show that if ( X k ) k∈N is a sequence of elements of an Q an p -definable family of subsets of Q m p which converges in the Hausdroff topology to X⊆ Q m p then X is Q an p -definable and dimX≤ lim sup k→∞ dim X k .