Quantifier Elimination for Lexicographic Products of Ordered Abelian Groups

Minimal sumsets in finite solvable groups

Let G be an Abelian group of order n, let k 2 be an integer, and A1, . . ., A k be non-empty subsets of G.and maximal by inclusion if for any i {1, . . ., k} and x G \ Ai, the collection (A1, . . ., Ai-1, Ai {x}, Ai+1, . .In this work, we study the problem of the maximal value of k (G).In particular, the maximal value of k (Z d ) for the cyclic group Z d is determined.Upper and lower bounds for k (G) are obtained for the Abelian group G.The structure of maximal k-sum-free set by capacity (by inclusion) is described for an arbitrary cyclic group.

We consider the first-order theory whose language has as nonlogical symbols the constant symbols 0 and 1, the binary relation symbols = and This theory of integers under addition is commonly called the 'Presburger Arithmetic' and is known to be decidable for truth [Presburger (1929), Hilbert and Bernays (1968)]. We prove here that there exists a decision procedure for this theory, involving quantifier elimination, for which there is a superexponential upper bound on the size of formula produced when all variables have been eliminated.

Addition sets in a finite group

Maximal sum-free sets in finite Abelian groups, III

Craig interpolation has become a versatile tool in formal verification, used for instance to generate program assertions that serve as candidates for loop invariants. In this paper, we consider Craig interpolation for quantifier-free Presburger arithmetic (QFPA). Until recently, quantifier elimination was the only available interpolation method for this theory, which is, however, known to be potentially costly and inflexible. We introduce an interpolation approach based on a sequent calculus for QFPA that determines interpolants by annotating the steps of an unsatisfiability proof with partial interpolants. We prove our calculus to be sound and complete. We have extended the Princess theorem prover to generate interpolating proofs, and applied it to a large number of publicly available Presburger arithmetic benchmarks. The results document the robustness and efficiency of our interpolation procedure. Finally, we compare the procedure against alternative interpolation methods, both for QFPA and linear rational arithmetic.

A compactness argument in the additive theory and the polynomial method

In the literature one can find three quite different proofs of the completeness of the infinite-valued sentential calculus of Lukasiewicz [8]: (i). the syntactical proof of Rose and Rosser [7], using McNaughton’s theorem, (ii). the algebraic proof of Chang [1, 2], using quantifier elimination in the first-order theory of divisible totally ordered abelian groups, and (iii). the recent proof of Cignoli [3], using the representation of free latticeordered abelian groups.

Let Q be a quasigroup. For $$\alpha ,\beta \in S_Q$$ let $$Q_{\alpha ,\beta }$$ be the principal isotope $$x*y = \alpha (x)\beta (y)$$ . Put $$\mathbf a(Q)= |\{(x,y,z)\in Q^3;$$ $$x(yz)) = (xy)z\}|$$ and assume that $$|Q|=n$$ . Then $$\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})$$ , and for every $$\alpha \in S_Q$$ there is $$\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2$$ , where $$f_x=|\{y\in Q;$$ $$ y = \alpha (y)x\}|$$ . If G is a group and $$\alpha $$ is an orthomorphism, then $$\mathbf a(G_{\alpha ,\beta })=n^2$$ for every $$\beta \in S_Q$$ . A detailed case study of $$\mathbf a(G_{\alpha ,\beta })$$ is made for the situation when $$G = \mathbb Z_{2d}$$ , and both $$\alpha $$ and $$\beta $$ are “natural” near-orthomorphisms. Asymptotically, $$\mathbf a(G_{\alpha ,\beta })>3n$$ if G is an abelian group of order n. Computational results: $$\mathbf a(7) = 17$$ and $$\mathbf a(8) \le 21$$ , where $$\mathbf a(n) = \min \{\mathbf a(Q);$$ $$ |Q|=n\}$$ . There are also determined minimum values for $$\mathbf a(G_{\alpha ,\beta })$$ , G a group of order $$\le 8$$ .

Many of the sporadic simple groups possess highly symmetric generating sets which can often be used to construct the groups, and which carry much information about their subgroup structure. We give a survey of results obtained so far.

The homological functors of a group were first introduced in homotopy theory. Some of the homological functors including the nonabelian tensor square and the Schur multiplier of abelian groups of prime power order are determined in this paper. The nonabelian tensor square of a group G introduced by Brown and Loday in 1987 is a special case of the nonabelian tensor product. Meanwhile, the Schur multiplier of G is the second cohomology with integer coefficients is named after Issai Schur. The aims of this paper are to determine the nonabelian tensor square and the Schur multiplier of abelian groups of order p5, where p is an odd prime

Recently, it was proved that every p-Schur ring over an abelian group of order <TEX>$p^3$</TEX> is Schurian. In this paper, we prove that every commutative p-Schur ring over a non-abelian group of order <TEX>$p^3$</TEX> is Schurian.

New constructions of divisible designs

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension 22n+1 for n ≥ 2 over an algebraically closed field of characteristic 0 with the group of group-like elements isomorphic to \(\mathbb {Z}_{2^{n}}\times \mathbb {Z}_{2^{n}}\). Moreover we classify all such nonisomorphic Hopf algebras of dimension 32 and show that they are not twist-equivalent to each other. More generally, given an abelian group of order 2 m−1 we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension 2 m which have this group as their group of group-like elements.

Let G = (V, E) be a simple graph with n vertices, and let A be an abelian group of order n. Let f: V(G) → A be a bijection, and define the weight of a vertex x by w(x) = Σy ∈ N(x) f(y), where N(x) denotes the open neighborhood of x. We call f an A-distance magic labeling if all vertex weights are equal, and an A-distance antimagic labeling if the weights of distinct vertices are pairwise different. The prime graph of a commutative ring R is the simple graph with vertex set R in which two vertices x and y are adjacent if and only if xRy = {0}. In this paper, we investigate the existence of A-distance magic and A-distance antimagic labelings of the prime graph over the ring Zn for several values of n.