@ARTICLE{SURREALO.MIZ,
  AUTHOR = {P\k{a}k, Karol},
  TITLE = {Integration of Game Theoretic and Tree Theoretic Approaches to {C}onway Numbers},
  SECTION1 = {Preorder of Surreal Numbers},
  SECTION2 = {Equivalence Relation of Preorder},
  SECTION3 = {Representative of Equivalence Class With a Unique Set of Properties},
  DAY = {12},
  MONTH = {December},
  YEAR = {2023},
  ADDRESS1 = {Faculty of Computer Science\\University of Bia{\l}ystok\\ Konstantego Cio{\l}kowskiego 1M, 15-245 Bia{\l}ystok\\Poland},
  SUMMARY = {In this article, we develop our formalised concept of Conway numbers
  \cite{Conway:2001} as outlined in \cite{SURREAL0.ABS}, focusing on the pre-order properties and
  birthday arithmetic. First, we present a formal analysis of the contents
  of Section {\it Properties of Order and Equality} in Chapter 1 of John Conway's
  seminal book. This section focuses on the pre-order structure of Conway numbers.\par
  Then, using the developed concept of Conway numbers, we thoroughly analyse
  the properties of surreal birthday arithmetic. We prove the
  {\it The Simplicity Theorem}, which improves our understanding of this
  specific arithmetic. It also makes it easier to manipulate birthday
  numbers in the context of pre-ordering surreal numbers.\par
  In the final part, we select the representatives of the equivalence classes that are defined by
  the relation equivalence relation $\thickapprox$ on surreal numbers
  such that $x \thickapprox y$ iff $ x\leq y$ and $y \leq x$.
  Representatives have a minimum-birthday as well as
  minimal-birthday as well as the left and right components
  of each representative having the smallest cardinality and
  such representatives as members.\par
  The formalisation is mainly based on \cite{Conway:2001, Ehrlich}, but also
  uses selected ideas proposed in \cite{alabdullah,Alling,schleicher}.}}

@article{schleicher,
author = {Schleicher, Dierk and Stoll, Michael},
year = {2004},
title = {An Introduction to {C}onway's Games and Numbers},
volume = {6},
journal = {Moscow Mathematical Journal},
doi = {10.17323/1609-4514-2006-6-2-359-388}
}

@Article{alabdullah,
  author =       {Maan T. Alabdullah and Essam El-Seidy and Neveen S. Morcos},
  title =        {On Numbers and Games},
  journal =      {International Journal of Scientific \& Engineering Research},
  year =         {2022},
  OPTkey =       {},
  volume =       {11},
  OPTnumber =    {2},
  OPTpages =     {},
  month =        {February},
  OPTnote =      {},
  OPTannote =    {}
}

@BOOK{Alling,
  AUTHOR =       {Norman L. Alling},
  TITLE =        {Foundations of Analysis Over Surreal Number Fields},
  PUBLISHER =    {North-Holland},
  YEAR =         {1987},
  isbn={9780444702265},
  lccn={87006735},
   number={141},
  series={Annals of Discrete Mathematics},
}



