Abstract We investigate nodes (elements that are comparable with all other elements) and their role in the structural analysis of pseudo-hoops. First, we study idempotent nodes and present a complete representation of pseudo-hoops with finitely many idempotent nodes, based on their decomposition into ordinal sums of pseudo-hoops. Next, we examine ordinal sum irreducible pseudo-hoops and prove that every Wajsberg pseudo-hoop with a non-trivial node is linearly ordered. Moreover, we show that the set of nodes $\text{Nod}(\mathbf E)$ of the ordinal sum of any family of Wajsberg pseudo-hoops forms a subalgebra. We then use the set of regular elements of a bounded representable pseudo-hoop $\mathbf E$ and its relationship with $\text{Nod}(\mathbf E)$ to determine whether $\mathbf E$ is ordinal sum reducible or irreducible. As a consequence, we establish that every finite basic pseudo-hoop can be expressed as an ordinal sum of a finite family ${\mathbf E_{1},\ldots ,\mathbf E_{n}}$ for some $n\in \mathbb N$, where $\mathbf E_{1},\ldots ,\mathbf E_{n-1}$ are linear Wajsberg pseudo-hoops and $\mathbf E_{n}$ is ordinal sum irreducible.

In the Eighties G. Cattaneo contributed to Hilbert space quantum mechanics models using the family of splitting subspaces of an inner product space in order to find completeness criteria. In this paper we show how that theory was developed in the las

In the frames of quantum structures, states are a very important notion that model probability on an algebraic structure. Already G. Boole observed that to calculate probability of event of the structure M, it is important to know only which two events A and B we can add such that if C = A+B, and P is a probability, then P (A+B) = P (A) + P (B); and the operation + is a partial one on M. If M is a Boolean algebra, then A+B: = A∪B whenever A∩B = ∅ or equivalently, A ≤ B′. Such two events A and B are said to be mutually excluding or summable, [DvPu]. Therefore, the state or finitely additive state on an algebraic structure (M; +, ′ , 0, 1) is any mapping s: M → [0, 1] such that (i) s(1) = 1, and (ii) s(a + b) = s(a) + s(b) whenever a + b is defined in M. The unary operation ′ is an orthogonal complement or a negation. The basic task is how to define a state on an algebraic structure like (M;⊕,, ∗ , 0, 1) or (M;⊕,, −, ∼ , 0, 1), that is, how to derive a partial operation + from the ⊕,? In more complicated structures, like orthomodular lattices and effect algebras, the most important example is the system L(H) of all closed subspaces of a Hilbert space H or the system of all Hermitian operators E(H) that are between the zero operator and the identity operator. Here the σ-additive states are of the form sφ(M) = (PMφ, φ),M ∈ L(H), φ ∈ H, s(M) = i λisφ(M) = tr(TPM), M ∈ L(H).