This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices ${\mathbf L}$ easily reduces to the case when ${\mathbf L}$ is a subdirect product of two simple lattices ${\mathbf L}_1$ and ${\mathbf L}_2$. Our main result claims that such a lattice is locally order affine complete iff ${\mathbf L}_1$ and ${\mathbf L}_2$ are tolerance trivial and one of the following three cases occurs: 1) ${\mathbf L}={\mathbf L}_1\times {\mathbf L}_2$, 2) ${\mathbf L}$ is a maximal sublattice of the direct product, 3) ${\mathbf L}$ is the intersection of two maximal sublattices, one containing $\langle 0,1\rangle $ and the other $\langle 1,0\rangle $.
The birth of a child and transition into home ownership are markers of progression along a life course. Research shows that pathways to home ownership have become more diverse and deviate from the traditional pathway which was characterised by marriage followed by the birth of a child before entering home ownership. This study investigates the timing and order of the two interrelated events of birth of a child and the transition to home ownership in Australia. Using the Household, Income and Labour Dynamics in Australia panel survey, we apply a multi-process event history analysis for describing the timing of each event following the formation of a cohabiting relationship. The results suggest that the likelihood of birth increases with prior home ownership attainment but as time passes following the purchase of a home, the likelihood of birth decreases, similarly, the likelihood of home ownership attainment decreased with time following birth.
It is proved by an order theoretical and purely algebraic method that any order bounded orthosymmetric bilinear operator $b\colon E\times E\rightarrow F$ where $E$ and $F$ are Archimedean vector lattices is symmetric. This leads to a new and short proof of the commutativity of Archimedean almost $f$-algebras.
Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when it is homotopy equivalent to a sphere., Nela Milošević, Zoran Z. Petrović., and Obsahuje seznam literatury
Let X be a completely regular Hausdorff space, E a boundedly complete vector lattice, Cb (X) the space of all, bounded, real-valued continuous functions on X, F the algebra generated by the zero-sets of X, and µ: Cb (X) → E a positive linear map. First we give a new proof that µ extends to a unique, finitely additive measure µ: F → E + such that ν is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of E +-valued finitely additive measures on F are proved, which extend some known results. Also, under certain conditions, the well-known Alexandrov’s theorem about the convergent sequences of σ-additive measures is extended to the case of order convergence.
Let $\mu \colon FX \to X$ be a principal bundle of frames with the structure group ${\rm Gl}_{n}(\mathbb R)$. It is shown that the variational problem, defined by ${\rm Gl}_{n}(\mathbb R)$-invariant Lagrangian on $J^{r} FX$, can be equivalently studied on the associated space of connections with some compatibility condition, which gives us order reduction of the corresponding Euler-Lagrange equations.
Ordered prime spectra of Boolean products of bounded DRl-monoids are described by means of their decompositions to the prime spectra of the components.
The reverse Wiener index of a connected graph $G$ is defined as \[ \Lambda (G)=\frac {1}{2}n(n-1)d-W(G), \] where $n$ is the number of vertices, $d$ is the diameter, and $W(G)$ is the Wiener index (the sum of distances between all unordered pairs of vertices) of $G$. We determine the $n$-vertex non-starlike trees with the first four largest reverse Wiener indices for $n\ge 8$, and the $n$-vertex non-starlike non-caterpillar trees with the first four largest reverse Wiener indices for $n\ge 10$.