We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations, where all the defining relations are of the form $r=1$. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular, our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied, these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.

Shott (2022, American Antiquity 87:794–815) argues that making inferences from ceramic data requires first inferring use lives of vessels—something that is difficult to do. This comment argues that the problem of differential use life becomes more tractable if the assemblage, rather than the vessel, is the unit of analysis. Aside from empirical reasons, theoretical considerations also favor the assemblage as the appropriate unit.

This is an addendum to a recent paper by Zaïmi, Bertin and Aljouiee [‘On number fields without a unit primitive element’, Bull. Aust. Math. Soc.93 (2016), 420–432], giving the answer to a question asked in that paper, together with some historical connections.

This study investigated the effects of short-term subclinical Zn deficiency on exocrine pancreatic activity and changes in digestive capacity. A total of forty-eight weaned piglets were fed ad libitum a basal diet (maize and soyabean meal) with adequate Zn supply (88 mg Zn/kg diet) during a 2-week acclimatisation phase. Animals were then assigned to eight dietary treatment groups (n 6) according to a complete randomised block design considering litter, live weight and sex. All pigs were fed restrictively (450 g diet/d) the basal diet but with varying ZnSO4.7H2O additions, resulting in 28·1, 33·6, 38·8, 42·7, 47·5, 58·2, 67·8 and 88·0 mg Zn/kg diet for a total experimental period of 8 d. Pancreatic Zn concentrations and pancreatic activities of trypsin, chymotrypsin, carboxypeptidase A and B, elastase and α-amylase exhibited a broken-line response to stepwise reduction in dietary Zn by declining beneath thresholds of 39·0, 58·0, 58·0, 41·2, 47·5, 57·7 and 58·0 mg Zn/kg diet, respectively. Furthermore, carboxypeptidase B and α-amylase activities were significantly lower in samples with reduced pancreatic Zn contents. Coefficients of faecal digestibility of DM, crude protein, total lipids and crude ash responded similarly to pancreatic enzyme activities by declining below dietary thresholds of 54·7, 45·0, 46·9 and 58·2 mg Zn/kg diet, respectively. In conclusion, (1) subclinical Zn deficiency impaired pancreatic exocrine enzymes, (2) this response was connected to pancreatic Zn metabolism and (3) the decline in catalytic activity impaired faecal digestibility already after 1 week of insufficient alimentary Zn supply and very early before clinical deficiency symptoms arise.

We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$, with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$.

Let G be a group of odd order that contains a non-central element x whose order is either a prime p ≥ 5 or 3l, with l ≥ 2. Then, in , the group of units of ℤG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large integers m we have that 〈um, vm〉 = 〈um〉 ∗ 〈vm〉 ≌ ℤ ∗ ℤ

Soient ${{p}_{1}},\,{{p}_{2}},\,{{p}_{3}}$ et $q$ des nombres premiers distincts tels que ${{p}_{1}}\,\equiv \,{{p}_{2}}\,\equiv \,{{p}_{3}}\,\equiv \,-q\,\equiv \,1\,(\bmod \,4)$, $k=\mathbf{Q}(\sqrt{{{p}_{1}}},\sqrt{{{p}_{2}}},\sqrt{{{p}_{3}}},\sqrt{q})$ et $\text{C}{{\text{l}}_{2}}(k)$ le 2-groupe de classes de $k$. A. Fröhlich a démontré que $\text{C}{{\text{l}}_{2}}(k)$ n’est jamais trivial. Dans cet article, nous donnons une extension de ce résultat, en démontrant que le rang de $\text{C}{{\text{l}}_{2}}(k)$ est toujours supérieur ou égal à 2. Nous démontrons aussi, que la valeur 2 est optimale pour une famille infinie de corps $k$.

Classifying is a fundamental operation in the acquisition of knowledge. Taxonomic theory can help students of cognition, evolutionary psychology, ethology, anatomy, and sociobiology to avoid serious mistakes, both practical and theoretical. More positively, it helps in generating hypotheses useful to a wide range of disciplines. Composite wholes, such as species and societies, are “individuals” in the logical sense, and should not be treated as if they were classes. A group of analogous features is a natural kind, but a group of homologous features is not. Imposing hypotheses justified only on the basis of nominalist, realist, phenomenalist, or conceptualist metaphysics upon the neurophysiology of organisms or upon the causes of behavior exemplifies the “psychologist's fallacy” of William James. Levels should be distinguished from their members and from classes of levels, and the ontological status of entities ranked at levels should be made clear. It is important not to confuse such categories as substance and process with one another. Several genetical terms, such as “gene” and chromosome,” are even more equivocal than has been realized. Discussions about units of selection, behavior, and thinking suffer from the ambiguity of a “unit of” an entity. An important source of misunderstanding about natural selection is the habit of treating it as an “agent”: in an important sense, natural selection does not “act” at all.

Herstein showed that the conjugacy class of a non-central element in the multiplicative group of a division ring is infinite. We prove similar results for units in algebras and orders and give applications to group rings.

AMS 2000 Mathematics subject classification: Primary 16U60. Secondary 16H05; 16S34; 20F24; 20C05

Let $G$ be an arbitrary group and let $U$ be a subgroup of the normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$ as a subgroup of finite index, then $U\,=\,G$. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

Based on results obtained in [15], we construct groups of special S-units for function fields of characteristic p>0, and show that they satisfy Gras-type Conjectures. We use these results in order to give a new proof of Chinburg‘s Ω$_3$-Conjecture on the Galois module structure of the group of S-units, for cyclic extensions of prime degree of function fields.

We prove that a refinement of Stark‘s Conjecture formulated by Rubin in Ann. Inst Fourier 4 (1996) is true up to primes dividing the order of the Galois group, for finite, Abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions, a statement stronger than Rubin‘s holds true.

Let q = p if p is an odd prime, q = 4 if p = 2. Let ζq be any primitive q-th root of unity, and let . We study the family of polynomials where Rn(X) and Sn(X) are the polynomials in the expansion We show that for fixed n, Pn(X; a) is irreducible for all but finitely many a ∈ O, and for p = 3, we show that it is irreducible for all a ∈ O. The roots are all real and are permuted cyclically by a linear fractional transformation defined over the real pn-th cyclotomic field. From the roots we obtain a non-maximal set of independent units for the splitting field. In the last section we briefly treat extensions of our methods to composite p.