@MISC{SRINGS_3.MIZ,
  AUTHOR = {Endou, Noboru and Nakasho, Kazuhisa and Shidama, Yasunari},
  TITLE = {$\sigma$-ring and $\sigma$-algebra of Sets},
  SECTION1 = {Preliminaries},
  SECTION2 = {Classical Semi-ring, Ring and $\sigma$-ring of Sets},
  SECTION3 = {Semi-algebra, Algebra and $\sigma$-algebra of Sets},
  SECTION4 = {Mutual Relationships between $\sigma$-ring
    and $\sigma$-algebra of Sets},
  DAY = {18},
  MONTH = {February},
  YEAR = {2015},
  ADDRESS1 = {Gifu National College of Technology, Gifu, Japan},
  ADDRESS2 = {Shinshu University, Nagano, Japan},
  ADDRESS3 = {Shinshu University, Nagano, Japan},
  SUMMARY = {In this article, semi-ring and semi-algebra of sets are formalized
  so as to construct a measure of a given set in the next step.
  Although a semi-ring of sets has already been formalized in
  \cite{SRINGS_1.ABS}, that is, strictly speaking, a definition of
  a quasi semi-ring of sets suggested in the last few decades
  \cite{GOGUADZE:2003}. We adopt a classical definition of a semi-ring of
  sets here to avoid such a confusion. Ring of sets and algebra
  of sets have been formalized as \verb!non empty preBoolean! set
  \cite{FINSUB_1.ABS} and \verb!Field_Subset!
  \cite{PROB_1.ABS}, respectively.
  In the second section, definitions of a ring and a $\sigma$-ring of sets, which
  are based on a semi-ring and a ring of sets respectively, are formalized and
  their related theorems are proved.
  In the third section, definitions of an algebra and a $\sigma$-algebra of sets,
  which are based on a semi-algebra and an algebra of sets respectively, are
  formalized and their related theorems are proved.
  In the last section, mutual relationships between $\sigma$-ring and
  $\sigma$-algebra
  of sets are formalized and some related examples are given.},
  NOTE = {This work was supported by JSPS KAKENHI 23500029 and 22300285.}}

@ARTICLE{GOGUADZE:2003,
year={2003},
issn={0001-4346},
journal={Mathematical Notes},
volume={74},
issue={3-4},
doi={10.1023/A:1026102701631},
title={About the Notion of Semiring of Sets},
url={http://dx.doi.org/10.1023/A%3A1026102701631},
publisher={Kluwer Academic Publishers-Plenum Publishers},
keywords={semiring; partition of a set; continuation of a partition; multiplicative class of sets; product of partitions},
author={Goguadze, D.F.},
pages={346-368},
language={English}
}

