Volume : 6, Issue : 12, December - 2017

SEMIGROUP VARIETIES AND ITS GEOMETRIC INTERPRETATION

Dr. Pankaj Kumar Chaudhary, Dr. Jawahar Lal Chaudhary, Dr. Niranjan Kumar Mishra, Gita Sinha

Abstract :

<p>&nbsp;<b style="text-align: justify;"><span style="font-size:10.0pt;&#10;line-height:115%;font-family:&quot;Times New Roman&quot;,&quot;serif&quot;;mso-fareast-font-family:&#10;&quot;Times New Roman&quot;;letter-spacing:1.0pt">Our main aim is to derive the embedding properties of modular lattice within its variety, into an algeaic Lattice. We extend here that every lattice which is either algeaic modular spatial or bi-algeaic is strongly spatial. Hermann and Roddy derived that every modular lattice embeds into some algeaic and spatial lattice. We show here that every n-distributive lattice embeds within its variety. It is illustrated by an example that only those lattice with a least and greatest element can be embedded which is join semi-distributive. The main derivation is that for every positive integer n, every n-distributive lattice is embedded within its variety which generalises word problem derived by C. Herrmann. Herrmann, Pickering, and Roddy, proved that every modular lattice L embeds into some algeaic and spatial lattice </span></b><span style="text-align: justify; font-size: 11pt; line-height: 115%; font-family: Calii, sans-serif; position: relative; top: 4pt;"><img width="8" height="18" src="file:///C:/Users/WWJ-08/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" v:shapes="_x0000_i1025" alt="" /></span><b style="text-align: justify;"><span style="font-size:10.0pt;line-height:115%;font-family:&quot;Times New Roman&quot;,&quot;serif&quot;;&#10;mso-fareast-font-family:&quot;Times New Roman&quot;;letter-spacing:1.0pt">&nbsp;that satisfies the same identities as L.</span></b></p> <p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;text-align:&#10;justify"><b><span style="font-size:10.0pt;line-height:115%;font-family:&quot;Times New Roman&quot;,&quot;serif&quot;;&#10;mso-fareast-font-family:&quot;Times New Roman&quot;;letter-spacing:1.0pt"><o:p></o:p></span></b></p>