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In this paper, we characterize and classify the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of SL3(k) and SL4(k) where k is a finite field of odd characteristic. We provide some generalized results for SLn(k). [ABSTRACT FROM AUTHOR]

Weakly type B semi-groups are generalized inverse semigroups on semi-abundant semigroups. This paper studied the semi group by the method of idempotent. As a generalization, the notion of a weakly type B unary semi-group was introduced by an idempotent method. Some basic properties of such unary semi-group were given. Moreover, some equivalent conditions for an arbitrary unary semigroup to be a weakly left type B semi-group were obtained. Finally, a semi-lattice decomposition of a weakly left B semi-group was given, and some results were obtained. [ABSTRACT FROM AUTHOR]

A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) the number of isomorphism classes of semisymmetric idempotent Latin squares of order n equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order n+1, and (2) suppose A and B are totally symmetric Latin squares of order n≢0 mod3. If A and B are paratopic then A and B are isomorphic. [ABSTRACT FROM AUTHOR]

We introduce the class of linear groups that do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic.We show that groups in this class have good closure properties, in addition to having properties akin to non-positive curvature, which were proved in [6].We give examples of abstract groups lying in this class, but also show that Gersten's free by cyclic group does not. This implies that it has no faithful linear representation of any dimension over any field of positive characteristic, nor can it be embedded in any complex unitary group. [ABSTRACT FROM AUTHOR]

We consider unipotent, weakly cancellative, left cancellative, right cancellative, cancellative monoids and groups with the unary operation of mapping onto the identity element and bands with the identity mapping as unary operations. Then form Malcev products of each of the former with varieties of rectangular bands, semilattices, normal bands and all bands. Regarding the resulting semigroups with the induced unary operations, we characterize these classes, some of them we provide with a construction, and determine a basis for their implications, for they all turn out to be quasivarieties. We also determine their relationship which we clarify by an inclusion diagram. [ABSTRACT FROM AUTHOR]

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a semigroup. Elements $a,b$ of $S$ are $\widetilde{\mathscr{R}}$-related if they have the same idempotent left identities. Then $S$ is weakly left ample if (1) idempotents of $S$ commute, (2) $\widetilde{\mathscr{R}}$ is a left congruence, (3) for any $a \in S$, $a$ is $\widetilde{\mathscr{R}}$-related to a (unique) idempotent, say $a^+$, and (4) for any element $a$ and idempotent $e$ of $S$, $ae=(ae)^+a$. Elements $a,b$ of $S$ are $\mathscr{R}^*$-related if, for any $x,y \in S^1$, $xa=ya$ if and only if $xb=yb$. Then $S$ is left ample if it satisfies (1), (3) and (4) relative to $\mathscr{R}^*$ instead of $\widetilde{\mathscr{R}}$. Further, $S$ is (weakly) ample if it is both (weakly) left and right ample. We establish several characterizations of these classes of semigroups. For weakly left ample ones we provide a construction of all such semigroups with zero all of whose nonzero idempotents are primitive. Among characterizations of weakly ample semigroups figure (strong) semilattices of unipotent monoids, and among those for ample semigroups, (strong) semilattices of cancellative monoids. This describes the structure of these two classes of semigroups in an optimal way, while, for the ‘one-sided’ case, the problem of structure remains open. [ABSTRACT FROM PUBLISHER]

Abstract: Necessary and sufficient conditions are presented for a square matrix A over a general field F to be the product of two unipotent matrices of index 2. This generalizes a result established by Wang and Wu (1991) for the case where F is the complex field. [ABSTRACT FROM AUTHOR]

Let G be a semisimple algebraic group defined over an algebraically closed field K of good characteristic p>0. Let u be a unipotent element of G of order pt, for some t ∈ N. In this paper it is shown that u lies in a closed subgroup of G isomorphic to the it Witt group Wt(K), which is a t-dimensional connected abelian unipotent algebraic group. 2000 Mathematics Subject Classification: 20G15. [ABSTRACT FROM PUBLISHER]

Let $X=\operatorname{Spec} B$ be an affine variety over a field of arbitrary characteristic, and suppose that there exists an action of a unipotent group (possibly neither smooth nor connected). The fundamental results are as follows. (1) An algorithm for computing invariants is given, by means of introducing a degree in the ring of functions of the variety, relative to the action. Therefore an algorithmic construction of the quotient, in a certain open set, is obtained. In the case of a Galois extension, $k\hookrightarrow B=K$, which is cyclic of degree $p=\text{ch} (k)$ (that is, such that the unipotent group is $G={\Bbb Z}/p {\Bbb Z}$), an element of minimal degree becomes an Artin--Schreier radical, and the method for computing invariants gives, in particular, the expression for any element of $K$ in terms of these radicals, with an explicit formula. This replaces the well-known formula of Lagrange (which is valid only when the degree of the extension and the characteristic are relatively prime) in the case of an extension of degree $p=\text{ch}(k)$. (2) In this paper we give an effective construction of a stable open subset where there is a quotient. In this sense we obtain an algebraic local criterion for the existence of a quotient in a neighbourhood. It is proved (provided the variety is normal) that, in the following cases, such an open set is the greatest one that admits a quotient: \begin{enumerate} \item[(a)] when the action is such that the orbits have dimension less than or equal to 1 (arbitrary characteristic) and, in particular, for any action of the additive group $G_a$; \item[(b)] in characteristic 0, when the action is proper (obtained from the results of Fauntleroy) or the group is abelian. 1991 Mathematics Subject Classification: primary 14L30; secondary 14D25, 14D20. [ABSTRACT FROM AUTHOR]

It is now widely believed that mammary epithelial cell plasticity, an important physiological process during the stages of mammary gland development, is exploited by the malignant cells for their successful disease progression. Normal mammary epithelial cells are heterogeneous and organized in hierarchical fashion, in which the mammary stem cells (MaSC) lie at the apex with regenerative capacity as well as plasticity. Despite the fact that the majority of studies supported the existence of multipotent MaSCs giving rise to both basal and luminal lineages, others proposed lineage restricted unipotent MaSCs. Consistent with the notion, the latest research has suggested that although normal MaSC subsets mainly stay in a quiescent state, they differ in their reconstituting ability, spatial localization, and molecular and epigenetic signatures in response to physiological stimuli within the respective microenvironment during the stages of mammary gland development. In this review, we will focus on current research on the biology of normal mammary stem cells with an emphasis on properties of cellular plasticity, self-renewal and quiescence, as well as the role of the microenvironment in regulating these processes. This will include a discussion of normal breast stem cell heterogeneity, stem cell markers, and lineage tracing studies. [ABSTRACT FROM AUTHOR]