The monogenic free inverse semigroup $FI_1$ is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of $FI_1$ is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of $FI_1$ is finitely presented as a semigroup if and only if it is a finite semilattice.

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren [‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly 96(2) (1989), 133–137], to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the literature. Finally, we devote a separate section to the cyclic case.

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.

We prove a new irreducibility result for polynomials over ${\mathbb Q}$ and we use it to construct new infinite families of reciprocal monogenic quintinomials in ${\mathbb Z}[x]$ of degree $2^n$ .

Suppose that $f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$, with $n\geq 3$ and $1\leq m<n$, is irreducible over $\mathbb{Q}$. By explicitly calculating the discriminant of $f(x)$, we prove that, when $\gcd (n,mB)=C=1$, there exist infinitely many values of $A$ such that the set $\{1,\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{2},\ldots ,\unicode[STIX]{x1D703}^{n-1}\}$ is an integral basis for the ring of integers of $\mathbb{Q}(\unicode[STIX]{x1D703})$, where $f(\unicode[STIX]{x1D703})=0$.

A waterhemp population (McLean County resistant, MCR) from McLean County, Illinois is resistant to both mesotrione and atrazine by elevated rates of herbicide metabolism. Research was conducted to investigate the inheritance of these resistance traits. Resistant and sensitive plants were crossed to obtain reciprocal F1 populations, which were then used to create pseudo-F2 and backcross (to sensitive parent; BCS) populations. The various populations were evaluated with whole-plant herbicide efficacy studies in a greenhouse. The responses of the F1 populations to both mesotrione and atrazine were intermediate when compared with parental populations. In the case of atrazine, BCS and F2 populations segregated 1 : 1 and 1 : 3, respectively, for susceptibility (S) : resistance (R), at a dose that controlled the sensitive parent but not the F1 or resistant parent. For mesotrione, variability was observed within the F1 populations, suggesting that mesotrione resistance is multigenic and the resistant parents used in the cross were not homozygous at the resistance loci. Furthermore, at low mesotrione doses, more F2 plants survived than expected on the basis of a single-gene trait, whereas at high doses, fewer F2 plants survived than expected. Dry weight data confirmed the conclusions obtained from survival data. Specifically, atrazine responses segregated into two discrete classes (R and S) in both the F2 and BCS populations, whereas mesotrione responses showed continuous distributions of phenotypes in F2 and BCS populations. We conclude that metabolism-based atrazine resistance in MCR is conferred by a single major gene, whereas inheritance of mesotrione resistance in this population is complex.

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is again finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.

Monogenic disorders account for only a minority of strokes. Yet, they have been particularly helpful in exploring basic disease mechanisms. This article summarizes some recent data on monogenic stroke while focusing on two conditions: CADASIL, as a genetic variant of ischemic small vessel disease, and familial forms of cerebral amyloid angiopathy, which share many properties with sporadic disease.

The potential of plant genetics in the overall context of integrated pest control could be viewed simply as the potential of plant resistance in pest management. Genetic resistance in plants is one of the most effective and economic means of controlling plant pests. Resistant plants are the first line defence against pests. Successes in breeding for pest resistance depend upon the sound knowledge of the plant genes controlling resistance to pests.

Recent report on the existence of two nonatlelic and independent genes, Rac1 and Rac2 in cowpea cultivars ICV 10 and TVu 310, and ICV 11 and ICV 12, respectively, from the ICIPE has broadened the scope of cowpea breeding against aphid biotypes. Use of biotechnology as a plant breeding tool in transferring resistance genes into agronomically suitable cultivars appears to hold good promise for the future.