International Journal of Algebra and Computation

In order to study the axiomatization of the if-then-else construct over possibly non-halting programs and tests, this paper introduces the notion of $C$-sets by considering the tests from an abstract $C$-algebra. When the $C$-algebra is an ada, the axiomatization is shown to be complete by obtaining a subdirect representation of $C$-sets. Further, this paper considers the equality test with the if-then-else construct and gives a complete axiomatization through the notion of agreeable $C$-sets.

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\mathbb{P}=\{P_1,\dots ,P_m\}$. Let $H_1,H_2$ be subgroups of $G$ such that $H_1$ is relatively quasiconvex with respect to $\mathbb{P}$ and $H_2$ is not parabolic. Suppose that $H_2$ is elementwise conjugate into $H_1$. Then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. The minimal length of the conjugator can be estimated. In the case, where $G$ is a limit group, it is sufficient to assume only that $H_1$ is a finitely generated and $H_2$ is an arbitrary subgroup of $G$.

The Tarski number of a nonamenable group is the smallest number of pieces needed for a paradoxical decomposition of the group. Nonamenable groups of piecewise projective homeomorphisms were introduced in [N. Monod, Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. 110(12) (2013) 4524–4527], and nonamenable finitely presented groups of piecewise projective homeomorphisms were introduced in [Y. Lodha and J. T. Moore, A finitely presented non amenable group of piecewise projective homeomorphisms, Groups, Geom. Dyn. 10(1) (2016) 177–200]. These groups do not contain non-abelian free subgroups. In this paper, we prove that the Tarski number of all groups in both families is at most 25. In particular, we demonstrate the existence of a paradoxical decomposition with 25 pieces.

Our argument also applies to any group of piecewise projective homeomorphisms that contains as a subgroup the group of piecewise $PS{L}_2(\mathbb{Z})$ homeomorphisms of $ℝ$ with rational breakpoints and an affine map that is a not an integer translation.

In this note, we characterize the Hilbert regularity of the Stanley–Reisner ring $K[\mathrm{\Delta }]$ in terms of the $f$-vector and the $h$-vector of a simplicial complex $\mathrm{\Delta }$. We also compute the Hilbert regularity of a Gorenstein algebra.

Let $G$ be an almost simple group with socle $A_n$, the alternating group of degree $n$. We prove that there is a unit of order $pq$ in the integral group ring of $G$ if and only if there is an element of that order in $G$ provided $p$ and $q$ are primes greater than $\frac{n}{3}$. We combine this with some explicit computations to verify the prime graph question for all almost simple groups with socle $A_n$ if $n\le 17$.