Groups with a ternary equivalence relation

We consider in a group \((G,\cdot )\) the ternary relation

$$\begin{aligned} \kappa := \{(\alpha , \beta , \gamma ) \in G^3 \ | \ \alpha \cdot \beta ^{-1} \cdot \gamma = \gamma \cdot \beta ^{-1} \cdot \alpha \} \end{aligned}$$

and show that \(\kappa \) is a ternary equivalence relation if and only if the set \( \mathfrak Z \) of centralizers of the group G forms a fibration of G (cf. Theorems 2, 3). Therefore G can be provided with an incidence structure

$$\begin{aligned} \mathfrak G:= \{\gamma \cdot Z \ | \ \gamma \in G , Z \in \mathfrak Z(G) \}. \end{aligned}$$

We study the automorphism group of \((G,\kappa )\), i.e. all permutations \(\varphi \) of the set G such that \( (\alpha , \beta , \gamma ) \in \kappa \) implies \((\varphi (\alpha ),\varphi (\beta ),\varphi (\gamma ))\in \kappa \). We show \(\mathrm{Aut}(G,\kappa )=\mathrm{Aut}(G,\mathfrak G)\), \(\mathrm{Aut} (G,\cdot ) \subseteq \mathrm{Aut}(G,\kappa )\) and if \( \varphi \in \mathrm{Aut}(G,\kappa )\) with \(\varphi (1)=1\) and \(\varphi (\xi ^{-1})= (\varphi (\xi ))^{-1}\) for all \(\xi \in G\) then \(\varphi \) is an automorphism of \((G,\cdot )\). This allows us to prove a representation theorem of \(\mathrm{Aut}(G,\kappa )\) (cf. Theorem 6) and that for \(\alpha \in G \) the maps

$$\begin{aligned} \tilde{\alpha }\ : \ G \rightarrow G;~ \xi \mapsto \alpha \cdot \xi ^{-1} \cdot \alpha \end{aligned}$$

of the corresponding reflection structure \((G, \widetilde{G})\) (with \( \tilde{G} := \{\tilde{\gamma }\ | \ \gamma \in G \}\)) are point reflections. If \((G ,\cdot )\) is uniquely 2-divisible and if for \(\alpha \in G\), \(\alpha ^{1\over 2}\) denotes the unique solution of \(\xi ^2=\alpha \) then with \(\alpha \odot \beta := \alpha ^{1\over 2} \cdot \beta \cdot \alpha ^{1\over 2}\), the pair \((G,\odot )\) is a K-loop (cf. Theorem 5).

     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

Ramanujan-type congruences for overpartitions modulo 16

Let \(\bar{p}(n)\) denote the number of overpartitions of \(n\). Recently, Fortin–Jacob–Mathieu and Hirschhorn–Sellers independently obtained 2-, 3- and 4-dissections of the generating function for \(\bar{p}(n)\) and derived a number of congruences for \(\bar{p}(n)\) modulo 4, 8 and 64 including \(\bar{p}(8n+7)\equiv 0 \pmod {64}\) for \(n\ge 0\). In this paper, we give a 16-dissection of the generating function for \(\bar{p}(n)\) modulo 16 and show that \(\bar{p}(16n+14)\equiv 0\pmod {16}\) for \(n\ge 0\). Moreover, using the \(2\)-adic expansion of the generating function for \(\bar{p}(n)\) according to Mahlburg, we obtain that \(\bar{p}(\ell ^2n+r\ell )\equiv 0\pmod {16}\), where \(n\ge 0\), \(\ell \equiv -1\pmod {8}\) is an odd prime and \(r\) is a positive integer with \(\ell \not \mid r\). In particular, for \(\ell =7\) and \(n\ge 0\), we get \(\bar{p}(49n+7)\equiv 0\pmod {16}\) and \(\bar{p}(49n+14)\equiv 0\pmod {16}\). We also find four congruence relations: \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n) \pmod {16}\) for \(n\ge 0\), \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {32}\) where \(n\) is not a square of an odd positive integer, \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {64}\) for \(n\not \equiv 1,2,5\pmod {8}\) and \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {128}\) for \(n\equiv 0\pmod {4}\).     

Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over \(\mathbb {Q}\), \(N_L\) the level of L, and \(\mathscr {M}(L)\)  be the linear space of \(\theta \)-series attached to the distinct classes in the genus of L. We prove that, for an odd prime \(p|N_L\), if \(L_p=L_{p,1}\,\bot \, L_{p,2}\), where \(L_{p,1}\) is unimodular, \(L_{p,2}\) is (p)-modular, and \(\mathbb {Q}_pL_{p,2}\) is anisotropic, then \(\mathscr {M}(L;p):=\) \(\mathscr {M}(L)\) \(+T_{p^2}.\) \(\mathscr {M}(L)\)  is stable under the Hecke operator \(T_{p^2}\). If \(L_2\) is isometric to \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) with \(\varepsilon \in \mathbb {Z}_2^{\times }\) and \(\kappa :=\frac{k-1}{2}\), then \(\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)\) is stable under the Hecke operator \(T_{2^2}\). Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.     

The deformation of symplectic critical surfaces in a Kähler surface-II—compactness

In this paper we consider the compactness of \(\beta \)-symplectic critical surfaces in a Kähler surface. Let M be a compact Kähler surface and \(\Sigma _i\subset M\) be a sequence of closed \(\beta _i\)-symplectic critical surfaces with \(\beta _i\rightarrow \beta _0\in (0,\infty )\). Suppose the quantity \(\int _{\Sigma _i}\frac{1}{\cos ^q\alpha _i}d\mu _i\) (for some \(q>4\)) and the genus of \(\Sigma _{i}\) are bounded, then there exists a finite set of points \({{\mathcal {S}}}\subset M\) and a subsequence \(\Sigma _{i'}\) which converges uniformly in the \(C^l\) topology (for any \(l<\infty \)) on compact subsets of \(M\backslash {{\mathcal {S}}}\) to a \(\beta _0\)-symplectic critical surface \(\Sigma \subset M\), each connected component of \(\Sigma \setminus {{\mathcal {S}}}\) can be extended smoothly across \({{\mathcal {S}}}\).     

On the image,characterization, and automatic continuity of $$\varvec{(\sigma }, \varvec{\tau }$$ )-derivations

The first main theorem of this paper asserts that any \((\sigma , \tau )\)-derivation d, under certain conditions, either is a \(\sigma \)-derivation or is a scalar multiple of (\(\sigma - \tau \)), i.e. \(d = \lambda (\sigma - \tau )\) for some \(\lambda \in \mathbb {C} \backslash \{0\}\). By using this characterization, we achieve a result concerning the automatic continuity of \((\sigma , \tau \))-derivations on Banach algebras which reads as follows. Let \(\mathcal {A}\) be a unital, commutative, semi-simple Banach algebra, and let \(\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}\) be two distinct endomorphisms such that \(\varphi \sigma (\mathbf e )\) and \(\varphi \tau (\mathbf e )\) are non-zero complex numbers for all \(\varphi \in \Phi _\mathcal {A}\). If \(d : \mathcal {A} \rightarrow \mathcal {A}\) is a \((\sigma , \tau )\)-derivation such that \(\varphi d\) is a non-zero linear functional for every \(\varphi \in \Phi _\mathcal {A}\), then d is automatically continuous. As another objective of this research, we prove that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every Jordan \(\sigma \)-derivation \(d:\mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.     

Infiltration Equation with Degeneracy on the Boundary

This paper is mainly about the infiltration equation

$$ {u_{t}}= \operatorname{div} \bigl(a(x)|u|^{\alpha }{ \vert { \nabla u} \vert ^{p-2}}\nabla u\bigr),\quad (x,t) \in \Omega \times (0,T), $$

where \(p>1\), \(\alpha >0\), \(a(x)\in C^{1}(\overline{\Omega })\), \(a(x)\geq 0\) with \(a(x)|_{x\in \partial \Omega }=0\). If there is a constant \(\beta \) such that \(\int_{\Omega }a^{-\beta }(x)dx\leq c\), \(p>1+\frac{1}{\beta }\), then the weak solution is smooth enough to define the trace on the boundary, the stability of the weak solutions can be proved as usual. Meanwhile, if for any \(\beta >\frac{1}{p-1}\), \(\int_{\Omega }a^{-\beta }(x)dxdt=\infty \), then the weak solution lacks the regularity to define the trace on the boundary. The main innovation of this paper is to introduce a new kind of the weak solutions. By these new definitions of the weak solutions, one can study the stability of the weak solutions without any boundary value condition.

     

Constructive Approximation in de Branges–Rovnyak Spaces

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\). We show that this is not the case for the de Branges–Rovnyak spaces \(\mathcal{H}(b)\). More precisely, we exhibit a space \(\mathcal{H}(b)\) in which the polynomials are dense and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). On the positive side, we prove the following approximation theorem for Toeplitz operators on general de Branges–Rovnyak spaces \(\mathcal{H}(b)\). If \((h_n)\) is a sequence in \(H^\infty \) such that \(\Vert h_n\Vert _{H^\infty }\le 1\) and \(h_n(0)\rightarrow 1\), then \(\Vert T_{\overline{h}_n}f-f\Vert _{\mathcal{H}(b)}\rightarrow 0\) for all \(f\in \mathcal{H}(b)\). Using this result, we give the first constructive proof that, if b is a nonextreme point of the unit ball of \(H^\infty \), then the polynomials are dense in \(\mathcal{H}(b)\).     

Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$-modules

In this paper, we show that for a positive operator A on a Hilbert \(C^*\)-module \( \mathscr {E} \), the range \( \mathscr {R}(A) \) of A is closed if and only if \( \mathscr {R}(A^\alpha ) \) is closed for all \(\alpha \in (0,1)\cup (1,+\,\infty )\), and this occurs if and only if \( \mathscr {R}(A)=\mathscr {R}(A^\alpha ) \) for all \(\alpha \in (0,1)\cup (1,+\,\infty )\). As an application, we prove that for an adjontable operator A if \(\mathscr {R}(A)\) is nonclosed, then \(\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty \). Finally, we show that for an adjointable operator A if \( \overline{\mathscr {R}(A^*) } \) is orthogonally complemented in \( \mathscr {E} \), then under certain coditions there exists an idempotent C and a unique operator X such that \( XAX=X, AXA=CA, AX=C \) and \( XA=P_{A^*} \), where \( P_{A^*} \) is the orthogonal projection of \( \mathscr {E} \) onto \( \overline{\mathscr {R}(A^*)}\).     

Periodic solutions for some double-delayed differential equations

We prove the existence of positive \(\omega \)-periodic solutions for the double-delayed differential equation

$$\begin{aligned} x^{\prime }(t)-a(t)g(x(t))x(t)=-\lambda (b(t)f(x(t-\tau (t))+c(t)h(x(t-\nu (t))), \end{aligned}$$

where \(\lambda \) is a positive parameter, \(a,b,c,\tau ,\nu \in C(\mathbb {R}, \mathbb {R})\) are \(\omega \)-periodic functions with \(a,b\ge 0,a,b\not \equiv 0,f,g,h\in C([0,\infty ),\mathbb {R})\) with \(g>0\) on \((0,\infty ),\) \(\ h\) is bounded, f is either superlinear or sublinear at \(\infty \) and could change sign.

     

Characterization of some derivations on von Neumann algebras via left centralizers

Let \(\mathfrak {M}\) be a von Neumann algebra, and let \(\mathfrak {T}:\mathfrak {M} \rightarrow \mathfrak {M}\) be a bounded linear map satisfying \(\mathfrak {T}(P^{2}) = \mathfrak {T}(P)P + \Psi (P,P)\) for each projection P of \(\mathfrak {M}\), where \(\Psi :\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is a bi-linear map. If \(\Psi \) is a bounded l-semi Hochschild 2-cocycle, then \(\mathfrak {T}\) is a left centralizer associated with \(\Psi \). By applying this conclusion, we offer a characterization of left \(\sigma \)-centralizers, generalized derivations and generalized \(\sigma \)-derivations on von Neumann algebras. Moreover, it is proved that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every bi-\(\sigma \)-derivation \(D:\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.     

Erratum to: Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams

A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume \(\Vert M\Vert \) of M is equal to \(\mathrm{Vol}(M)/v_n\), where \(v_n\) is a constant depending only on the dimension of M. The same result also holds for complete finite-volume hyperbolic manifolds without boundary, while Jungreis proved that the ratio \(\mathrm{Vol}(M)/\Vert M\Vert \) is strictly smaller than \(v_n\) if M is compact with nonempty geodesic boundary. We prove here a quantitative version of Jungreis’ result for \(n\ge 4\), which bounds from below the ratio \(\Vert M\Vert /\mathrm{Vol}(M)\) in terms of the ratio \(\mathrm{Vol}(\partial M)/\mathrm{Vol}(M)\). As a consequence, we show that, for \(n\ge 4\), a sequence \(\{M_i\}\) of compact hyperbolic n-manifolds with geodesic boundary satisfies \(\lim _i \mathrm{Vol}(M_i)/\Vert M_i\Vert =v_n\) if and only if \(\lim _i \mathrm{Vol}(\partial M_i)/\mathrm{Vol}(M_i)=0\). We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension 3.     

Some new Kirkman signal sets

A decomposition of the blocks of an \(\textsf {STS}(v)\) into partial parallel classes of size m is equivalent to a Kirkman signal set \(\textsf {KSS}(v,m)\). We give decompositions of \(\textsf {STS}(4v-3)\) into classes of size \(v-1\) when \(v \equiv 3 \pmod {6}\), \(v \not = 3\). We also give decompositions of \(\textsf {STS}(v)\) into classes of various sizes when v is a product of two arbitrary integers that are both congruent to \(3 \pmod {6}\). These results produce new families of \(\textsf {KSS}(v,m)\).     

Space–Time Fractional Equations and the Related Stable Processes at Random Time

In this paper, we consider the general space–time fractional equation of the form \(\sum _{j=1}^m \lambda _j \frac{\partial ^{\nu _j}}{\partial t^{\nu _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)\), for \(\nu _j \in \left( 0,1 \right] \) and \(\beta \in \left( 0,1 \right] \) with initial condition \(w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)\). We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), where \(\varvec{S}_n^{2\beta }\) is an isotropic stable process independent from \(\mathcal {L}^{\nu _1, \ldots , \nu _m}(t)\), which is the inverse of \(\mathcal {H}^{\nu _1, \ldots , \nu _m} (t) = \sum _{j=1}^m \lambda _j^{1/\nu _j} H^{\nu _j} (t)\), \(t>0\), with \(H^{\nu _j}(t)\) independent, positively skewed stable random variables of order \(\nu _j\). The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), supplies a probabilistic representation for the solutions of the fractional equations above and coincides for \(\beta = 1\) with the n-dimensional Brownian motion at the random time \(\mathcal {L}^{\nu _1, \ldots , \nu _m} (t)\), \(t>0\). The iterated process \(\mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t)\), \(t>0\), inverse to \(\mathfrak {H}^{\nu _1, \ldots , \nu _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/\nu _j} \, _1H^{\nu _j} \left( \, _2H^{\nu _j} \left( \, _3H^{\nu _j} \left( \ldots \, _{r}H^{\nu _j} (t) \ldots \right) \right) \right) \), \(t>0\), permits us to construct the process \(\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t) \right) \), \(t>0\), the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For \(r \rightarrow \infty \) and \(\beta = 1\), we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation \(\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)\). Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.     

On Hecke groups,Schwarzian triangle functions and a class of hyper-elliptic functions

Let m be a positive integer \(\ge \)3 and \(\lambda =2\cos \frac{\pi }{m}\). The Hecke group \(\mathfrak {G}(\lambda )\) is generated by the fractional linear transformations \(\tau + \lambda \) and \(-\frac{1}{\tau }\) for \(\tau \) in the upper half plane \(\mathbb H\) of the complex plane \(\mathbb C\). We consider a set of functions \(\mathfrak {f}_0, \mathfrak {f}_i\) and \(\mathfrak {f}_{\infty }\) automorphic with respect to \(\mathfrak {G}(\lambda )\), constructed from the conformal mapping of the fundamental domain of \(\mathfrak {G}(\lambda )\) to the upper half plane \(\mathbb H\), and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of \(\lambda =1\) and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of \(\pi \) in terms of the hypergeometric series.     

Negative Ricci curvature on some non-solvable Lie groups

We show that for any non-trivial representation \((V, \pi )\) of \(\mathfrak {u}(2)\) with the center acting as multiples of the identity, the semidirect product \(\mathfrak {u}(2) \ltimes _\pi V\) admits a metric with negative Ricci curvature that can be explicitly obtained. It is proved that \(\mathfrak {u}(2) \ltimes _\pi V\) degenerates to a solvable Lie algebra that admits a metric with negative Ricci curvature. An n-dimensional Lie group with compact Levi factor \(\mathrm {SU}(2)\) admitting a left invariant metric with negative Ricci is therefore obtained for any \(n \ge 7\).     

On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals \(\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}\), \(u \in \mathbf{I}^d\), \(d \ge 1\), the normalized partial sums \(\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )\), \(t \in [0,1]\) and its polygonal version \({S}_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta } \xrightarrow {\mathbb {P}} \theta \), and weaker moment conditions (\(p = 2\) if \(d = 1\)) are assumed.     

Chains,antichains, and complements in infinite partition lattices

We consider the partition lattice \(\Pi (\lambda )\) on any set of transfinite cardinality \(\lambda \) and properties of \(\Pi (\lambda )\) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly \(\lambda \); (II) there are maximal chains in \(\Pi (\lambda )\) of cardinality \(> \lambda \); (III) a regular cardinal \(\lambda \) is strongly inaccessible if and only if every maximal chain in \(\Pi (\lambda )\) has size at least \(\lambda \); if \(\lambda \) is a singular cardinal and \(\mu ^{< \kappa } < \lambda \le \mu ^\kappa \) for some cardinals \(\kappa \) and (possibly finite) \(\mu \), then there is a maximal chain of size \(< \lambda \) in \(\Pi (\lambda )\); (IV) every non-trivial maximal antichain in \(\Pi (\lambda )\) has cardinality between \(\lambda \) and \(2^{\lambda }\), and these bounds are realised. Moreover, there are maximal antichains of cardinality \(\max (\lambda , 2^{\kappa })\) for any \(\kappa \le \lambda \); (V) all cardinals of the form \(\lambda ^\kappa \) with \(0 \le \kappa \le \lambda \) occur as the cardinalities of sets of complements to some partition \(\mathcal {P} \in \Pi (\lambda )\), and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation.