The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

For two given $\omega$-terms $\alpha$ and $\beta$, the word problem for $\omega$-terms over a variety $\boldsymbol{\mathrm{V}}$ asks whether $\alpha=\beta$ in all monoids in $\boldsymbol{\mathrm{V}}$. We show that the word problem for $\omega$-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by $\omega$-terms (this property is also known as $\omega$-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable

On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids

This paper investigates the class of finitely presented monoids defined by homogeneous (length-preserving) relations from a computational perspective. The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being biautomatic are investigated for this class of monoids. The first main result shows that for any consistent combination of these properties and their negations, there is a homogeneous monoid with exactly this combination of properties. We then introduce the new concept of abstract Rees-commensurability (an analogue of the notion of abstract commensurability for groups) in order to extend this result to show that the same statement holds even if one restricts attention to the class of n-ary homogeneous monoids (where every side of every relation has fixed length n). We then introduce a new encoding technique that allows us to extend the result partially to the class of n-ary multihomogenous monoids

On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids

This paper investigates the class of finitely presented monoids defined by homogeneous (length-preserving) relations from a computational perspective. The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being biautomatic are investigated for this class of monoids. The first main result shows that for any consistent combination of these properties and their negations, there is a homogeneous monoid with exactly this combination of properties. We then introduce the new concept of abstract Rees-commensurability (an analogue of the notion of abstract commensurability for groups) in order to extend this result to show that the same statement holds even if one restricts attention to the class of n-ary homogeneous monoids (where every side of every relation has fixed length n). We then introduce a new encoding technique that allows us to extend the result partially to the class of n-ary multihomogenous monoids

Serum Calcium Levels Are Associated with Novel Cardiometabolic Risk Factors in the Population-Based CoLaus Study

BACKGROUND: Associations of serum calcium levels with the metabolic syndrome and other novel cardio-metabolic risk factors not classically included in the metabolic syndrome, such as those involved in oxidative stress, are largely unexplored. We analyzed the association of albumin-corrected serum calcium levels with conventional and non-conventional cardio-metabolic risk factors in a general adult population. METHODOLOGY/PRINCIPAL FINDINGS: The CoLaus study is a population-based study including Caucasians from Lausanne, Switzerland. The metabolic syndrome was defined using the Adult Treatment Panel III criteria. Non-conventional cardio-metabolic risk factors considered included: fat mass, leptin, LDL particle size, apolipoprotein B, fasting insulin, adiponectin, ultrasensitive CRP, serum uric acid, homocysteine, and gamma-glutamyltransferase. We used adjusted standardized multivariable regression to compare the association of each cardio-metabolic risk factor with albumin-corrected serum calcium. We assessed associations of albumin-corrected serum calcium with the cumulative number of non-conventional cardio-metabolic risk factors. We analyzed 4,231 subjects aged 35 to 75 years. Corrected serum calcium increased with both the number of the metabolic syndrome components and the number of non-conventional cardio-metabolic risk factors, independently of the metabolic syndrome and BMI. Among conventional and non-conventional cardio-metabolic risk factors, the strongest positive associations were found for factors related to oxidative stress (uric acid, homocysteine and gamma-glutamyltransferase). Adiponectin had the strongest negative association with corrected serum calcium. CONCLUSIONS/SIGNIFICANCE: Serum calcium was associated with the metabolic syndrome and with non-conventional cardio-metabolic risk factors independently of the metabolic syndrome. Associations with uric acid, homocysteine and gamma-glutamyltransferase were the strongest. These novel findings suggest that serum calcium levels may be associated with cardiovascular risk via oxidative stress