Elliptic Reflection Structures, K-Loop Derivations and Triangle-Inequality

This paper is a part of our general aim to study properties of elliptic and ordered elliptic geometries and then using some of these properties to introduce new concepts and develop their theories. If ${(P,\mathfrak{G}, \equiv,\tau)}$ denotes an elliptic geometry ordered via a separation ?? then there are polar points o and ?? and on the line ${ \overline{K} := \overline{\infty,o}}$ there can be established an operation ??+?? such that ${(\overline{K},+)}$ becomes a commutative group and the map ${ a^+ :\overline{K}\to \overline{K} ; x \mapsto a + x}$ is a motion on ${\overline{K}}$ . The separation ?? induces on ${\overline{K}}$ a cyclic order ?? with [o, e, ??] = 1 such that ${(\overline{K},+,\omega)}$ becomes a cyclic ordered group. For ${a,b \in K := \overline{K} {\setminus}\{\infty\}}$ we set ${a < b :\Longleftrightarrow [a,b,\infty] =1}$ and for all ${a\in K\,a < \infty}$ . Then (K,?<) is a totally ordered set. We show there is a surjective distance function: $$ \lambda : P \times P \to \overline{K}_+ := \{x \in \overline{K}\,|\,o \leq x\leq\infty\}, $$ with ?? ${\lambda(a,b) = \lambda(c,d) \ \Longleftrightarrow (a,b) \equiv (c,d)}$ ??. We prove in the first part of our project like (cf. Gr?ger in Mitt Math Ges Hamburg 11:441?C457, 1987) the following triangle-inequality: (cf. Theorem 8.2). If (a, b, c) is a triangle consisting of pairwise not polar points with ??(a, c), ??(b, c) < e then ??(a, b) ?? ??(a, c) + ??(b, c) < ??.     

Group inverse for a class of 2×2 anti-triangular block matrices over skew fields

Suppose K is a skew field. Let K ^m×n denote the set of all m×n matrices over K. In this paper, we give necessary and sufficient conditions for the existence and explicit representations of the group inverses of the block matrices in the following three cases, respectively:

1. $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)$ ;
2. $\mathrm{rank}(S)=\mathrm{rank}(AB^{\pi})$ ;
3. $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)=\mathrm{rank}(AB^{\pi})$ ,

where A,B,C??K ^n×n , B ^# exists, R(B)=R(C), N(B)=N(C) and S=B ^?? AB ^?? . The paper??s conclusions generalized some related results of Zhao and Bu (Electron. J. Linear Algebra 21:63?C75, 2010).     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

Boundary partial regularity for a class of biharmonic maps

We consider the Dirichlet problem for biharmonic maps u from a bounded, smooth domain ${\Omega\subset\mathbb R^n (n\ge 5)}$ to a compact, smooth Riemannian manifold ${N\subset{\mathbb {R}}^l}$ without boundary. For any smooth boundary data, we show that if u is a stationary biharmonic map that satisfies a certain boundary monotonicity inequality, then there exists a closed subset ${\Sigma\subset\overline{\Omega}}$ , with ${H^{n-4}(\Sigma)=0}$ , such that ${\displaystyle u\in C^\infty(\overline\Omega\setminus\Sigma, N)}$ .     

A class of {\Sigma _{3}^{0}} modular lattices embeddable as principal filters in {\mathcal{L}^{\ast }(V_{\infty })}

Let I ₀ be a a computable basis of the fully effective vector space V _∞ over the computable field F. Let I be a quasimaximal subset of I ₀ that is the intersection of n maximal subsets of the same 1-degree up to *. We prove that the principal filter ${\mathcal{L}^{\ast}(V,\uparrow )}$ of V = cl(I) is isomorphic to the lattice ${\mathcal{L}(n, \overline{F})}$ of subspaces of an n-dimensional space over ${\overline{F}}$ , a ${\Sigma _{3}^{0}}$ extension of F. As a corollary of this and the main result of Dimitrov (Math Log 43:415–424, 2004) we prove that any finite product of the lattices ${(\mathcal{L}(n_{i}, \overline{F }_{i}))_{i=1}^{k}}$ is isomorphic to a principal filter of ${\mathcal{ L}^{\ast}(V_{\infty})}$ . We thus answer Question 5.3 “What are the principal filters of ${\mathcal{L}^{\ast}(V_{\infty}) ?}$ ” posed by Downey and Remmel (Computable algebras and closure systems: coding properties, handbook of recursive mathematics, vol 2, pp 977–1039, Stud Log Found Math, vol 139, North-Holland, Amsterdam, 1998) for spaces that are closures of quasimaximal sets.     

Regularized Trace of the Cauchy Transform

We give an exact value of the regularized trace of Cauchy transform C _?? i.e. the sum of the series ${\sum\nolimits_{n=1}^{\infty}( s_{n}^{2}( C_{\Omega}) -\frac{\vert \Omega \vert}{\pi n})}$ .     

Differentiable vectors and unitary representations of Fréchet–Lie supergroups

A locally convex Lie group G has the Trotter property if, for every $x_1, x_2 \in \mathfrak{g }$ , $$\begin{aligned} \exp _G(t(x_1 + x_2))=\lim _{n \rightarrow \infty } \left(\exp _G\left(\frac{t}{n}x_1\right)\exp _G\left(\frac{t}{n}x_2\right)\right)^n \end{aligned}$$ holds uniformly on compact subsets of $\mathbb{R }$ . All locally exponential Lie groups have this property, but also groups of automorphisms of principal bundles over compact smooth manifolds. A key result of the present article is that, if G has the Trotter property, $\pi : G \rightarrow {\mathrm{GL}}(V)$ is a continuous representation of G on a locally convex space, and $v \in V$ is a vector such that $\overline{\mathtt{d}\pi }(x)v :=\frac{d}{dt}|_{t=0} \pi (\exp _G(tx))v$ exists for every $x \in \mathfrak{g }$ , then the map $\mathfrak{g }\rightarrow V,x \mapsto \overline{\mathtt{d}\pi }(x)v$ is linear. Using this result we conclude that, for a representation of a locally exponential Fréchet–Lie group G on a metrizable locally convex space, the space of $\mathcal{C }^{k}$ -vectors coincides with the common domain of the k-fold products of the operators $\overline{\mathtt{d}\pi }(x)$ . For unitary representations on Hilbert spaces, the assumption of local exponentiality can be weakened to the Trotter property. As an application, we show that for smooth (resp., analytic) unitary representations of Fréchet–Lie supergroups $(G,\mathfrak{g })$ where G has the Trotter property, the common domain of the operators of $\mathfrak{g }=\mathfrak{g }_{\overline{0}}\oplus \mathfrak{g }_{\overline{1}}$ can always be extended to the space of smooth (resp., analytic) vectors for G.     

On the asymptotic behavior of solutions of Emden–Fowler equations on time scales

Consider the Emden-Fowler dynamic equation $$ x^{\Delta\Delta}(t)+p(t)x^\alpha(t)=0,\:\:\alpha >0 , \qquad \qquad \qquad \qquad (0.1) $$ where ${p\in C_{rd}([t_0,\infty)_{\mathbb{T}},\mathbb{R}), \alpha}$ is the quotient of odd positive integers, and ${\mathbb{T}}$ denotes a time scale which is unbounded above and satisfies an additional condition (C) given below. We prove that if ${\int^\infty_{t_0}t^\alpha |p(t)|\Delta t<\infty}$ (and when ???=?1 we also assume lim _t???? tp(t)??(t)?=?0), then (0.1) has a solution x(t) with the property that $$ \lim_{t\rightarrow\infty} \frac{x(t)}{t}=A\neq 0.$$      

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Beyond the Trudinger-Moser supremum

Let Ω be a bounded, smooth domain in ${\mathbb{R}^2}$ . We consider the functional $$I(u) = \int_\Omega e^{u^2}\,dx$$ in the supercritical Trudinger-Moser regime, i.e. for ${\int_\Omega |\nabla u|^2dx > 4\pi}$ . More precisely, we are looking for critical points of I(u) in the class of functions ${u \in H_0^1 (\Omega )}$ such that ${\int_\Omega |\nabla u|^2 \, dx = 4\, \pi \, k\, (1+\alpha)}$ , for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with ${\int_\Omega |\nabla u|^2dx = 4\pi(1 + \alpha)}$ for any bounded domain Ω, 2-peak critical points with ${\int_\Omega |\nabla u|^2dx = 8\pi(1 + \alpha)}$ for non-simply connected domains Ω, and k-peak critical points with ${\int_\Omega |\nabla u|^2 dx = 4k \pi(1 + \alpha)}$ if Ω is an annulus.     

Uniformly Quasiregular Maps on the Compactified Heisenberg Group

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification $\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}$ of the Heisenberg group ?¹ equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere ${\mathbb{S}}^{n} $ was proven by Martin (Conform. Geom. Dyn. 1:24?C27, 1997). Moreover, we construct uniformly quasiregular mappings on $\bar{ {\mathbb{H}}}^{1}$ with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on $\bar{ {\mathbb{H}}}^{1}$ there exists a measurable CR structure ?? which is equivariant under the semigroup ?? generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.     

Immunity properties and strong positive reducibilities

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has ${\overline{K}\not\le_{\rm ss} B}$ (respectively, ${\overline{K}\not\le_{\overline{\rm s}} B}$ ): here ${\le_{\overline{\rm s}}}$ is the finite-branch version of s-reducibility, ??_ss is the computably bounded version of ${\le_{\overline{\rm s}}}$ , and ${\overline{K}}$ is the complement of the halting set. Restriction to ${\Sigma^0_2}$ sets provides a similar characterization of the ${\Sigma^0_2}$ hyperhyperimmune sets in terms of s-reducibility. We also show that no ${A \geq_{\overline{\rm s}}\overline{K}}$ is hyperhyperimmune. As a consequence, ${\deg_{\rm s}(\overline{K})}$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.     

Uniqueness of diffusion operators and capacity estimates

Let Ω be a connected open subset of R ^d . We analyse L ₁-uniqueness of real second-order partial differential operators ${H = - \sum^d_{k,l=1} \partial_k c_{kl} \partial_l}$ and ${K = H + \sum^d_{k=1}c_k \partial_k + c_0}$ on Ω where ${c_{kl} = c_{lk} \in W^{1,\infty}_{\rm loc}(\Omega), c_k \in L_{\infty,{\rm loc}}(\Omega), c_0 \in L_{2,{\rm loc}}(\Omega)}$ and C(x) = (c _kl (x)) > 0 for all ${x \in \Omega}$ . Boundedness properties of the coefficients are expressed indirectly in terms of the balls B(r) associated with the Riemannian metric C ^?1 and their Lebesgue measure |B(r)|. First, we establish that if the balls B(r) are bounded, the Täcklind condition ${\int^\infty_R dr r({\rm log}|B(r)|)^{-1} = \infty}$ is satisfied for all large R and H is Markov unique then H is L ₁-unique. If, in addition, ${C(x) \geq \kappa (c^{T} \otimes c)(x)}$ for some ${\kappa > 0}$ and almost all ${x \in \Omega}$ , ${{\rm div} c \in L_{\infty,{\rm loc}}(\Omega)}$ is upper semi-bounded and c ₀ is lower semi-bounded, then K is also L ₁-unique. Secondly, if the c _kl extend continuously to functions which are locally bounded on ?Ω and if the balls B(r) are bounded, we characterize Markov uniqueness of H in terms of local capacity estimates and boundary capacity estimates. For example, H is Markov unique if and only if for each bounded subset A of ${\overline\Omega}$ there exist ${\eta_n \in C_c^\infty(\Omega)}$ satisfying , where ${\Gamma(\eta_n) = \sum^d_{k,l=1}c_{kl} (\partial_k \eta_n) (\partial_l \eta_n)}$ , and for each ${\varphi \in L_2(\Omega)}$ or if and only if cap(?Ω) = 0.     

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

An extension of Mercer’s theory to L p

Let X be a topological space, either locally compact or first countable, endowed with a strictly positive measure ?? and ${\mathcal{K}:L^2(X,\nu)\to L^2(X,\nu)}$ an integral operator generated by a Mercer like kernel K. In this paper we extend Mercer??s theory for K and ${\mathcal{K}}$ under the assumption that the function ${x\in X\to K(x,x)}$ belongs to some L ^p/2(X, ??), p??? 1. In particular, we obtain series representations for K and some powers of ${\mathcal{K}}$ , with convergence in the p-mean, and show that the range of certain powers of ${\mathcal{K}}$ contains continuous functions only. These results are used to estimate the approximation numbers of a modified version of ${\mathcal{K}}$ acting on L ^p (X, ??).     

Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions

We consider the nonlinear delay differential evolution equation $$\left\{\begin{array}{ll} u'(t) \in Au(t) + f(t, u_t), \quad \quad t \in \mathbb{R}_+,\\ u(t) = g(u)(t),\qquad \qquad \quad t \in [-\tau, 0], \end{array} \right.$$ u ′ ( t ) ∈ A u ( t ) + f ( t , u t ) , t ∈ R + , u ( t ) = g ( u ) ( t ) , t ∈ [ - τ , 0 ] , where τ ≥ 0, X is a real Banach space, A is the infinitesimal generator of a nonlinear semigroup of contractions whose Lipschitz seminorm decays exponentially as ${t \mapsto {\rm{e}}^{-\omega t}}$ t ? e - ω t when ${t \to + \infty}$ t → + ∞ and ${f : {\mathbb{R}}_+ \times C([-\tau, 0]; \overline{D(A)}) \to X}$ f : R + × C ( [ - τ , 0 ] ; D ( A ) ¯ ) → X is jointly continuous. We prove that if f Lipschitz with respect to its second argument and its Lipschitz constant ? satisfies the condition ${\ell{\rm{e}}^{\omega\tau} < \omega, g : C_b([-\tau, +\infty); \overline{D(A)}) \to C([-\tau, 0]; \overline{D(A)})}$ ? e ω τ < ω , g : C b ( [ - τ , + ∞ ) ; D ( A ) ¯ ) → C ( [ - τ , 0 ] ; D ( A ) ¯ ) is nonexpansive and (I – A)^?1 is compact, then the unique C ⁰-solution of the problem above is almost periodic.     

On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).     

Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.     

The Robin and Wentzell-Robin Laplacians on Lipschitz Domains

Let $\Omega\subset{\Bbb R}^N$ be a bounded domain with Lipschitz boundary. We prove in the first part that a realization of the Laplacian with Robin boundary conditions $\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ generates a holomorphic $C_0$ -semigroup of angle $\pi/2$ on $C(\overline{\Omega})$ if $0<\beta_0\le \beta\in L^{\infty}(\partial \Omega)$ . With the same assumption on $\Omega$ and assuming that $0\le\beta\in L^{\infty}(\partial \Omega)$ , we show in the second part that one can define a realization of the Laplacian on $C(\overline{\Omega})$ with Wentzell-Robin boundary conditions $\Delta u+\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ and this operator generates a $C_0$ -semigroup.