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Let $({𝔄}_n{,}^{\rho })$ be the involution monoid of Annular monoid ${𝔄}_n$ under the rotation involution ${}^{\rho }$. The involution monoids $({𝔄}_1{,}^{\rho })$ and $({𝔄}_2{,}^{\rho })$ are easily seen to be finitely based; Auinger et al. proved that $({𝔄}_n{,}^{\rho })$ is inherently non-finitely based if $n\ge 4$. In this paper, we show that $({𝔄}_3{,}^{\rho })$ is finitely based by providing a finite identity basis for $({𝔄}_3{,}^{\rho })$, which answers an open question posed by Auinger et al. Therefore, the involution monoid $({𝔄}_n{,}^{\rho })$ is finitely based if and only if $n\le 3$.

An infinite word w avoids a pattern p with the involution if there is no substitution for the variables in p and no involution on substituted variables such that the resulting word is a factor of w. An avoidance index of pattern p is the smallest alphabet size for which a word exists such that p is avoided. A pattern is called unary, if only one variable occurs in it. In this paper, we give the avoidance indices for all unary patterns with involution.

This paper presents a systematic study for abstract Banach measure algebras over homogeneous spaces of compact groups. Let $H$ be a closed subgroup of a compact group $G$ and $G/H$ be the left coset space associated to the subgroup $H$ in $G$. Also, let $M(G/H)$ be the Banach measure space consists of all complex measures over $G/H$. Then we introduce the abstract notions of convolution and involution over the Banach measure space $M(G/H)$.

Let $A$ be a superalgebra with superinvolution or graded involution over a field of characteristic zero and let ${\chi }_{n_1,\dots ,n_4}(A)$, $n_1+\cdots +n_4=n$, be the $(n_1,\dots ,n_4)$-cocharacter of $A$. The ∗-colengths sequence, $l_n^{\ast }(A)$, $n=1,2,\dots$, is the sum of the multiplicities in the decomposition of the $(n_1,\dots ,n_4)$-cocharacter ${\chi }_{n_1,\dots ,n_4}(A)$, for all $n=n_1+\cdots +n_4\ge 1$. The main purpose of this paper is to classify the superalgebras with superinvolution with ∗-colengths sequence bounded by three. Moreover, we shall extend to the general case, the analogous result proved by do Nascimento and Vieira in [Superalgebras with graded involution and star-graded colength bounded by 3, Linear Multilinear Algebra 67(10) (2019) 1999–2020] for finite-dimensional superalgebras with graded involution.

It is known, since the early 2000s, that the Catalan monoid $C_n$ generated by $n$ elements is finitely based if and only if $n\le 3$. The main goal of this paper is to prove that the involution monoid $(C_3,{}^{*})$ is non-finitely based. Therefore, in contrast, combining with previous results yields that the involution Catalan monoid $(C_n,{}^{*})$ is finitely based if and only if $n=1$.

The Kiselman monoid $K_n$ generated by $n$ elements is also considered. Although the semigroups $C_n$ and $K_n$ have recently been shown to satisfy the same identities, it is unknown if the same result holds when they are considered as involution semigroups. Nevertheless, it is deduced from the main result that the involution Kiselman monoid $(K_n,{}^{*})$ is finitely based if and only if $n=1$.

Let $S_g$ be a closed oriented surface of genus $g$ and let $\text{Mod}(S_g)$ be the mapping class group. When the genus is at least 3, $\text{Mod}(S_g)$ can be generated by torsion elements. We prove the following results: For $g\ge 4$, $\text{Mod}(S_g)$ can be generated by four torsion elements. Three generators are involutions and the fourth one is an order three element. $\text{Mod}(S_3)$ can be generated by five torsion elements. Four generators are involutions and the fifth one is an order three element.

A $2n$-component non-splittable spatial graph $G=G_1\cup \cdots \cup G_{2n}$ in $S^3$ possibly admits a smooth involution on its exterior whose fixed point set is a closed surface $F$ of genus $g\ge 1$. It is known that $n\ge g$ holds if $G$ is a graph link in $S^3$. In this paper, we consider $G$ to be a hyperbolic spatial graph in a closed orientable 3-manifold $M$. We first study the case where $M$ is the 3-sphere, and provide a method for constructing $G$ and $F$ with $a_1+\cdots +a_n>g>1$, where $1-a_i$ is the Euler characteristic of $G_i$ for $1\le i\le n$. Next, we study the case where $M$ is a closed orientable 3-manifold in relation to Heegaard splittings and Dehn surgeries.

Let R be a commutative ring, G a group and RG its group ring. Let φ: RG → RG denote the R-linear extension of an involution φ defined on G. An element x in RG is said to be antisymmetric if φ(x) = -x. A characterization is given of when the antisymmetric elements of RG commute except when Char(R) = 3.

Let * denote the canonical involution of the group algebra KG induced by the map x ↦ x^-1 for x ∈ G. In case K is a real extension of ℚ, we consider Cayley unitary elements built out of skew elements k = α(x - x^-1) in KG such that 1 + k is invertible in KG, for α ∈ K and x ∈ G. The constructions involve an interesting sequence in the coefficients of (1 + k)^-1 which is the Fibonacci sequence when α = 1.

It is presented a complete categorical description of one of the main differentiation algorithms for representations of posets with involution — Differentiation II — constructed originally by the second author at the end of the 80's on the base of the matrix approach. A similar categorical description of some simpler additional differentiations is given as well.

We classify the abelian groups G, for which the following property holds: for every subgroup H, every ϕ ∈ Aut (H) has an extension ψ ∈ Aut(G). We also classify the infinite polycyclic-by-finite groups and the finite nilpotent 2′-groups having this property. Fuchs, Bertholf, Walls and Tomkinson did similar work for groups which have the property that homomorphisms of its subgroups extend to the whole group.

Let R be a commutative ring, G a group and RG its group ring. Let φ : RG → RG denote the R-linear extension of an involution φ defined on G. An element x in RG is said to be φ-antisymmetric if φ (x) = -x. A characterization is given of when the φ-antisymmetric elements of RG commute. This is a completion of earlier work.

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)_* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)_* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)_* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈log₂ t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring.

Let R be a prime ring, which is not commutative, with involution * and with Q_ms(R) the maximal symmetric ring of quotients of R. An additive map δ : R → R is called a Jordan *-derivation if δ(x²) = δ(x)x* + xδ(x) for all x ∈ R. A Jordan *-derivation of R is called X-inner if it is of the form x ↦ xa - ax* for x ∈ R, where a ∈ Q_ms(R). We prove that any Jordan *-derivation of R is X-inner if char R ≠ 2 or deg(S(R)) > 4, where S(R) := {x ∈ R|x* = x}.

Cartan–Brauer–Hua Theorem is a well-known theorem which states that if $R$ is a subdivision ring of a division ring $D$ which is invariant under all elements of $D$ or $dR{d}^{-1}\subseteq R$ for all $d\in D\setminus \{0\}$, then either $R=D$ or $R$ is contained in the center of $D$. The invariance idea of this basic theorem is the main notion of this paper. We prove that if $D$ is a division ring with involution ${}^{\ast }$ and $M$ is a subspace of $D$ which is invariant under all symmetric elements of $D$, then either $M$ is contained in the center of $D$ or is a Lie ideal of $D$. Also, we show that if $T$ is a self-invariant subfield of a non-commutative division ring $D$ with a nontrivial automorphism, then $T$ contains at least one non-central proper subfield of $D$.

Let $G$ be a finite group and $I(G)$ be the set of the elements $x$ of $G$ such that $x^2=1$. In this paper, we characterize all groups $G$ such that $|I(G)|\ge \frac{\left|G\right|}{3}$ and $G$ is not a 2-group.

In this paper, we introduce and study a new exactness property in the sense of categorical algebra, which can be seen as a natural strengthening of the well-known Mal’tsev property. A variety of universal algebras has this exactness property if and only if its algebraic theory contains binary terms $a_1,\dots ,a_n$ and an $(n+1)$-ary term $d$ satisfying $d(a_1(x,y),\dots ,a_n(x,y),y)=x$, and $a_j(x,y)=a_j(y,x)$ for each $j\in \{1,\dots ,n\}$. We also study the “weak” version of this exactness property, which similarly strengthens the “weak Mal’tsev property” in the sense of the second author.

The aim of this paper is to give a complete description of ∗-mappings. Indeed, we define and study a more general class of semiderivations (respectively generalized semiderivations), that we call ∗-semiderivations (respectively ∗-generalized semiderivations). In particular, we prove that for the prime rings with involution, these new definitions coincide with the classical definitions of semiderivations and generalized semiderivations, respectively.

The problem of whether a metabolic idempotent of a central simple algebra with involution is contained in an invariant quaternion subalgebra is investigated. As an application, the similar problem is studied for skew-symmetric elements whose squares lie in the square of the underlying field.