If $R$ is a commutative ring with identity and $\le$ is defined by letting $a\le b$ mean $ab=a$ or $a=b$, then $(R,\le )$ is a partially ordered ring. Necessary and sufficient conditions on $R$ are given for $(R,\le )$ to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings $Z_n$ of integers mod $n$ for $n\ge 2$. In particular, if $R$ is reduced, then $(R,\le )$ is a lattice iff $R$ is a weak Baer ring, and $(R,\le )$ is a distributive lattice iff $R$ is a Boolean ring, $Z_3,Z_4$, $Z_2\left[x\right]/x^2{Z}_2\left[x\right]$, or a four element field.

Effect algebras are very natural logical structures as carriers of probabilities and states. They were introduced for modeling of sets of propositions, properties, questions, or events with fuzziness, uncertainty or unsharpness. Nevertheless, there are effect algebras without any state, and questions about the existence (for non-modular) are still unanswered. We show that every Archimedean atomic lattice effect algebra with at most five blocks (maximal MV-subalgebras) has at least one state, which...

We prove the $*$–version of the Joly–Becker theorem: a skew field admits a $*$–ordering of level $n$ iff it admits a $*$–ordering of level $n\ell$ for some (resp. all) odd $\ell \in \mathbb{N}$. For skew fields with an imaginary unit and fields stronger results are given: a skew field with imaginary unit that admits a $*$–ordering of higher level also admits a $*$–ordering of level $1$. Every field that admits a $*$–ordering of higher level admits a $*$–ordering of level $1$ or $2$

The main result of the paper characterizes continuous local derivations on a class of commutative Banach algebra $A$ that all of its squares are positive and satisfying the following property: Every continuous bilinear map $\Phi$ from $A\times A$ into an arbitrary Banach space $B$ such that $\Phi (a,b)=0$ whenever $ab=0$, satisfies the condition $\Phi (ab,c)=\Phi (a,bc)$ for all $a,b,c\in A$.

Let $R$ be an Archimedean partially ordered ring in which the square of every element is positive, and $N\left(R\right)$ the set of all nilpotent elements of $R$. It is shown that $N\left(R\right)$ is the unique nil radical of $R$, and that $N\left(R\right)$ is locally nilpotent and even nilpotent with exponent at most $3$ when $R$ is 2-torsion-free. $R$ is without non-zero nilpotents if and only if it is 2-torsion-free and has zero annihilator. The results are applied on partially ordered rings in which every element $a$ is expressed as $a=a_1-a_2$ with positive $a_1$,...

Let $A$ and $B$ be two Archimedean vector lattices and let ${\left(A^{\text{'}}\right)}_n^{\text{'}}$ and ${\left(B^{\text{'}}\right)}_n^{\text{'}}$ be their order continuous order biduals. If $\Psi :A\times A\to B$ is a positive orthosymmetric bimorphism, then the triadjoint ${\Psi }^{***}:{\left(A^{\text{'}}\right)}_n^{\text{'}}\times {\left(A^{\text{'}}\right)}_n^{\text{'}}\to {\left(B^{\text{'}}\right)}_n^{\text{'}}$ of $\Psi$ is inevitably orthosymmetric. This leads to a new and short proof of the commutativity of almost $f$-algebras.