Relatively free dimonoids and bar-units

This paper is devoted to the study of the problem of adjoining a set of bar-units to dimonoids. We give necessary and sufficient conditions for adjoining a set of bar-units to the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), and prove that it is impossible to adjoin a set of bar-units to the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid. As consequences, we establish that it is impossible to extend by a set of bar-units the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid to a generalized digroup. We also count the cardinalities of the free left [Formula: see text]-dinilpotent dimonoid and the free [Formula: see text]-dimonoid for a finite case.

Structure of relatively free trioids

Loday and Ronco introduced the notions of a~trioid and a trialgebra, and constructed the free trioid of rank 1 and the free trialgebra. This paper is a survey of recent developments in the study of free objects in the varieties of trioids and trialgebras. We present the constructions of the free trialgebra and the free trioid, the free commutative trioid, the free n-nilpotent trioid, the free left (right) n-trinilpotent trioid, and the free rectangular trioid. Some of these results can be applied to constructing relatively free trialgebras.

A note on a generalization of injective modules

As a proper generalization of injective modules in term of supplements, we say that a module $M$ has the property (ME) if, whenever $M\subseteq N$, $M$ has a supplement $K$ in $N$, where $K$ has a mutual supplement in $N$. In this study, we obtain that $(1)$ a semisimple $R$-module $M$ has the property (E) if and only if $M$ has the property (ME); $(2)$ a semisimple left $R$-module $M$ over a commutative Noetherian ring $R$ has the property (ME) if and only if $M$ is algebraically compact if and only if almost all isotopic components of $M$ are zero; $(3)$ a module $M$ over a von Neumann regular ring has the property (ME) if and only if it is injective; $(4)$ a principal ideal domain $R$ is left perfect if every free left $R$-module has the property (ME)

Noncommutative inclusion–exclusion, representations of left regular bands and the Tsetlin library

We find a semigroup [Formula: see text], whose category of partial representations contains the representation category [Formula: see text] of the free left regular band [Formula: see text]. We use this to construct a resolution for the absolute kernel of a representation of [Formula: see text], for which the kernel [Formula: see text] of the Markov operation in the Tsetlin library model is a prominent example. We obtain a formula for the dimension of the absolute kernel, generalizing the equality of the dimension of [Formula: see text] to the number of derangements of order [Formula: see text].

Cardiac Phlegmon: Infectious Endocarditis Causing Ventricular Wall Rupture

Background Rupture of the cardiac ventricular wall is a rare and essentially lethal complication of infectious endocarditis valvularis. Case Description We report a case of a 49-year-old female patient with infectious endocarditis of the aortic valve. Following aortic valve replacement, the patient developed a sudden free left ventricular wall rupture. Immediate reoperation was successful. Histopathology revealed a myocardial infarction due to septic thromboembolism causing a phlegmonlike myocardial appearance. Conclusion This is a rare case of a myocardial phlegmon with subsequent cardiac lateral wall rupture in the context of an infectious endocarditis with septic coronary embolism.