Facebook Instagram Twitter RSS Feed PodBean Back to top on side

States with values in the Łukasiewicz groupoid

In: Mathematica Slovaca, vol. 66, no. 2
Milan Matoušek - Pavel Pták

Details:

Year, pages: 2016, 335 - 342
Keywords:
Łukasiewicz t-norm, Łukasiewicz groupoid, Boolean algebra, orthomodular poset, quantum logic, classical and $Z_2$-valued state, Greechie paste job
About article:
In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let $\ast$ stand for the Łukasiewicz t-norm on $[0,1]^2$. Let us consider the operation $\diamond$ on $[0,1]$ by setting $x\diamond y=(x^\perp\ast y^\perp)^\perp\ast(x\ast y)^\perp$, where $x^\perp = 1-x$. Let us call the triple $L=([0,1],\diamond,1)$ the Łukasiewicz groupoid. Let $B$ be a Boolean algebra. Denote by $\mathcal{L}(B)$ the set of all $L$-valued measures ($L$-valued states). We show as a main result of this paper that the family $\mathcal{L}(B)$ consists precisely of the union of classical real states and $Z_2$-valued states. With the help of this result we characterize the $L$-valued states on orthomodular posets. Since the orthomodular posets are often understood as ``quantum logics'' in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any $L$-valued state.
How to cite:
ISO 690:
Matoušek, M., Pták, P. 2016. States with values in the Łukasiewicz groupoid. In Mathematica Slovaca, vol. 66, no.2, pp. 335-342. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0139

APA:
Matoušek, M., Pták, P. (2016). States with values in the Łukasiewicz groupoid. Mathematica Slovaca, 66(2), 335-342. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0139
About edition:
Published: 1. 4. 2016