fa رنگ پذیری در توپولوژِی جبر جهانی Universal algebra علمی پژوهشی بنیادی S در این مطالعه هدف بیان نحوه رنگ آمیزی سطوح توپولوژِکی به صورتیکه رنگ &shy;ها دارای مرز اما بدون فاصله و با کمترین عدد رنگی است. اینکه یک سطح را می&shy;توان با حداقل چه تعداد رنگ رنگ آمیزی کرد به صورتیکه شرایط ایجاد تعریف یک نوع نگاشت با شرط بدون نقطه ثابت بودن را همراه داشته باشد. این نگاشت را نگاشت رنگی نامیده و در شرایط مختلف&nbsp; فضا مانند فشردگی یا پارافشردگی، نرمال یا متریک بودن و پیوستگی و... مورد بررسی و تحلیل قرار می&shy;گیرد و متناسب با نوع هر فضا خواص مربوط به نگاشت&shy;ها را تغییر داده تا نتیجه مورد نظر حاصل شود. در ادامه با اثبات قضایا و لم&shy;های متعدد، عدد&shy; رنگی منسوب به هر یک از نگاشت&shy;ها با شرایط مختص به آن را بدست آورده می شود. اثبات می شود که &nbsp;به جز یک استنثنا که در متن به آن اشاره شده است این عدد از حداقل <span dir="LTR">3</span> &nbsp;البته و بسته به &nbsp;شرایط خاص هر فضا تا حداکثر <span dir="LTR">n+3</span> افزایش می&shy;یابد. که <span dir="LTR">n</span> می تواند بسته به شرایط متناهی هم باشد. <div class="tlid-language-bar ls-wrap"> <div class="tl-wrap"> <div class="tl-sugg"><div class="sl-sugg-button-container"></div><div class="ls-right-arrow"></div></div> <div class="sugg-fade"></div><div class="tl-more tlid-open-target-language-list" aria-label="More" role="button" tabindex="0"></div></div> </div> <div class="source-wrap"> <div class="input-full-height-wrapper tlid-input-full-height-wrapper"> <div class="source-input"> <div class="tlid-input-area input-area less-padding" id="input-wrap"><span style="line-height:22.0pt"><b><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Introduction</span></span></b></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">The aim of this study, is painting of topological surfaces with the least number of colors without the distance, and the colors have a border. For this purpose, we need a color mapping. In this mapping, we have not any fixed point, and we can colorable the map with least colors. </span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">Definition: Let f</span><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:">∶</span></span><span style="font-size:10.0pt"> X &rarr; X be a graph without a fixed point. f is colorable with k colors, if there is C={C_1,&hellip;,C_K}, where all C_i do not include {(x, f(x)}. Or similarly, for every i=1,&hellip;, k, there is the equation C_i &cap; f(C_i )=</span><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:">&empty;</span></span><span style="font-size:10.0pt">.</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">Also, we define some concepts such as Compression, Metric, or non-Compression of space. Also, to achieve the desired result of each space, we change the properties of the maps.</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Material and methods</span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">In this work, first, we define the properties and conditions of the color mapping and color number. Also, by the study of properties of each space, we choose the best of space. One of the best conditions of this space is the lowest color number and higher efficiency. Finally, we proved &nbsp;that this number is finite, and we can do coloring space with some maps and conversely.</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Results and discussion</span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">In this work, we define the properties and conditions of the color mapping and color number. We presented some theorems and Lemma in the article and proved them for coloring of any space by coloring map, the coloring number</span> <span style="font-size:10.0pt">is at least 3 and at most is a n+3. Also, we proved the coloring number finite and we can &nbsp;do coloring space with some maps and conversely.</span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Conclusion</span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">The following conclusions were drawn from this research.</span></span></span></span> <ul> <li style="list-style-type:none"> <ul style="list-style-type:disc"> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">the coloring number is at least 3.</span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">the coloring number is at most n+3.</span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">coloring number is finite and we can do coloring space with some maps.</span></span></span></span></li> <li style="text-align:justify"><span style="font-size:11pt"><span style="line-height:20.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">We can do the coloring of any space by the finite coloring map.</span></span></span></span></li> </ul> </li> </ul> </div> </div> </div> </div> نگاشت‌های رنگی, عدد رنگی, پوشش نقطه ای, پیوستگی, همسانریختی Color mappings, color numbers, dot coverage, continuity, homogeneity 138 151 http://mmr.khu.ac.ir/browse.php?a_code=A-10-396-3&slc_lang=fa&sid=1 hmaid erfanianoraei dehrokhi حمید عرفانیان اورعی دهرخی hamiderfanian23@yahoo.com 10031947532846005471 10031947532846005471 Yes دانشگاه دامغان majid erfanian oraei مجید عرفانیان اورعی erfaniyan@uoz.ac.ir 10031947532846005472 10031947532846005472 No دانشگاه زاهدان