Multifraction reduction III: The case of interval monoids

We investigate gcd-monoids, which are cancellative monoids in which any two elements admit a left and a right gcd, and the associated reduction of multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the word problem for the enveloping group. Here we consider the particular case of interval monoids associated with finite posets. In this way, we construct gcd-monoids, in which reduction of multifractions has prescribed properties not yet known to be compatible: semi-convergence of reduction without convergence, semi-convergence up to some level but not beyond, non-embeddability into the enveloping group (a strong negation of semi-convergence).Comment: 23 pages ; v2 : cross-references updated ; v3 : one example added, typos corrected; final version due to appear in Journal of Combinatorial Algebr

The order topology for a von Neumann algebra

The order topology $\tau_o(P)$ (resp. the sequential order topology $\tau_{os}(P)$) on a poset $P$ is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra $M$ we consider the following three posets: the self-adjoint part $M_{sa}$, the self-adjoint part of the unit ball $M_{sa}^1$, and the projection lattice $P(M)$. We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology to the other standard locally convex topologies on $M$, and relate the properties of the order topology to the underlying operator-algebraic structure of $M$

Finite Size Scaling in 2d Causal Set Quantum Gravity

We study the $N$-dependent behaviour of $\mathrm{2d}$ causal set quantum gravity. This theory is known to exhibit a phase transition as the analytic continuation parameter $\beta$, akin to an inverse temperature, is varied. Using a scaling analysis we find that the asymptotic regime is reached at relatively small values of $N$. Focussing on the $\mathrm{2d}$ causal set action $S$, we find that $\beta \langle S\rangle$ scales like $N^\nu$ where the scaling exponent $\nu$ takes different values on either side of the phase transition. For $\beta > \beta_c$ we find that $\nu=2$ which is consistent with our analytic predictions for a non-continuum phase in the large $\beta$ regime. For $\beta<\beta_c$ we find that $\nu=0$, consistent with a continuum phase of constant negative curvature thus suggesting a dynamically generated cosmological constant. Moreover, we find strong evidence that the phase transition is first order. Our results strongly suggest that the asymptotic regime is reached in $\mathrm{2d}$ causal set quantum gravity for $N \gtrsim 65$.Comment: 32 pages, 27 figures (v2 typos and missing reference fixed