We deal with decomposition theorems for modular measures $\mu :L\to G$ defined on a D-lattice with values in a Dedekind complete $\ell$-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete $\ell$-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for $\ell$-group-valued...

Using the general hypergraph technique developed in [7], we first give a much simpler proof of Shultz's theorem [10]: Each compact convex set is affinely homeomorphic to the state space of an orthomodular lattice. We also present partial solutions to open questions formulated in [10] - we show that not every compact convex set has to be a state space of a unital orthomodular lattice and that for unital orthomodular lattices the state space characterization can be obtained in the context of unital...

For a class $K$ of structures and $A\in K$ let ${Con}^{*}\left(A\right)$ resp. ${Con}^K\left(A\right)$ denote the lattices of $*$-congruences resp. $K$-congruences of $A$, cf. Weaver [25]. Let ${Con}^{*}\left(K\right):=I\{{Con}^{*}\left(A\right):A\in K\}$ where $I$ is the operator of forming isomorphic copies, and ${Con}^r\left(K\right):=I\{{Con}^K\left(A\right):A\in K\}$. For an ordered algebra $A$ the lattice of order congruences of $A$ is denoted by ${Con}^{<}\left(A\right)$, and let ${Con}^{<}\left(K\right):=I\{{Con}^{<}\left(A\right):A\in K\}$ if $K$ is a class of ordered algebras. The operators of forming subdirect squares and direct products are denoted by $Q^s$ and $P$, respectively. Let $\lambda$ be a lattice identity and let $\Sigma$ be a set of lattice identities. Let $\Sigma {\vDash }_c\lambda (r;Q^s,P)$ denote...

In this paper some results on direct summands of Goldie extending elements are studied in a modular lattice. An element $a$ of a lattice $L$ with $0$ is said to be a Goldie extending element if and only if for every $b\le a$ there exists a direct summand $c$ of $a$ such that $b\wedge c$ is essential in both $b$ and $c$. Some characterizations of decomposition of a Goldie extending element in a modular lattice are obtained.

We present a construction of finite distributive lattices with a given skeleton. In the case of an H-irreducible skeleton K the construction provides all finite distributive lattices based on K, in particular the minimal one.