Minimal conditions on Clifford semigroup congruences

A known result in groups concerning the inheritance of minimal conditions on normal subgroups by subgroups with finite indexes is extended to semilattices of groups [E(S),Se,ϕe,f] with identities in which all ϕe,f are epimorphisms (called q partial groups). Formulation of this result in terms of q congruences is also obtained

MINIMAL CONDITIONS ON CLIFFORD SEMIGROUP CONGRUENCES

A known result in groups concerning the inheritance of minimal conditions on normal subgroups by subgroups with finite indexes is extended to semilattices of groups [E(S), S e , ϕ e, f ] with identities in which all ϕ e, f are epimorphisms (called q partial groups). Formulation of this result in terms of q congruences is also obtained

On semilattices of groups whose arrows are epimorphisms

A q partial group is defined to be a partial group, that is, a strong semilattice of groups S=[E(S);Se,ϕe,f] such that S has an identity 1 and ϕ1,e is an epimorphism for all e∈E(S). Every partial group S with identity contains a unique maximal q partial group Q(S) such that (Q(S))1=S1. This Q operation is proved to commute with Cartesian products and preserve normality. With Q extended to idempotent separating congruences on S, it is proved that Q(ρK)=ρQ(K) for every normal K in S. Proper q partial groups are defined in such a way that associated to any group G, there is a proper q partial group P(G) with (P(G))1=G. It is proved that a q partial group S is proper if and only if S≅P(S1) and hence that if S is any partial group, there exists a group M such that S is embedded in P(M). P epimorphisms of proper q partial groups are defined with which the category of proper q partial groups is proved to be equivalent to the category of groups and epimorphisms of groups

Physicians' guideline adherence is associated with long-term heart failure mortality in outpatients with heart failure with reduced ejection fraction: the QUALIFY international registry

Background: Physicians' adherence to guideline-recommended therapy is associated with short-term clinical outcomes in heart failure (HF) with reduced ejection fraction (HFrEF). However, its impact on longer-term outcomes is poorly documented. Here, we present results from the 18-month follow-up of the QUALIFY registry. Methods and results: Data at 18 months were available for 6118 ambulatory HFrEF patients from this international prospective observational survey. Adherence was measured as a continuous variable, ranging from 0 to 1, and was assessed for five classes of recommended HF medications and dosages. Most deaths were cardiovascular (CV) (228/394) and HF-related (191/394) and the same was true for unplanned hospitalizations (1175 CV and 861 HF-related hospitalizations, out of a total of 1541). According to univariable analysis, CV and HF deaths were significantly associated with physician adherence to guidelines. In multivariable analysis, HF death was associated with adherence level [subdistribution hazard ratio (SHR) 0.93, 95% confidence interval (CI) 0.87–0.99 per 0.1 unit adherence level increase; P = 0.034] as was composite of HF hospitalization or CV death (SHR 0.97, 95% CI 0.94–0.99 per 0.1 unit adherence level increase; P = 0.043), whereas unplanned all-cause, CV or HF hospitalizations were not (all-cause: SHR 0.99, 95% CI 0.9–1.02; CV: SHR 0.98, 95% CI 0.96–1.01; and HF: SHR 0.99, 95% CI 0.96–1.02 per 0.1 unit change in adherence score; P = 0.52, P = 0.2, and P = 0.4, respectively). Conclusion: These results suggest that physicians' adherence to guideline-recommended HF therapies is associated with improved outcomes in HFrEF. Practical strategies should be established to improve physicians' adherence to guidelines. © 2019 The Authors. European Journal of Heart Failure © 2019 European Society of Cardiolog