States on non-commutative residuated lattices

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K00024
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Theory of informatics
• Research Institution: Tokyo Denki University
• Principal Investigator: KONDO Michiro 東京電機大学, システムデザイン工学部, 教授 (40211916)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: residuated lattice / state

Outline of Final Research Achievements

In this research programme, I consider some properties of states on non-commutative residuated lattices and prove that for any non-commutative residuated lattice L, if there exists a state s on L, then the quotient structure L/ker(s) by a kernel ker(s) of the state s is a (commutative) MV-algebra. Therefore, it follows from this result that the measurement problems in quantum logics reduce to those of MV-algebras. After that, I aslo consider algebraic properties of residuated lattices L with state operators $\sigma$ which are not outer languages like mappings but inner ones like modal operators. These structures (L,$\sigma$) are called $\sigma$-residuated lattices. I got some results about algebraic properties of $\sigma$-residuated lattices and published a paper in a journal.

Free Research Field

数理論理学