Geometric Lattice Structure of Covering-Based Rough Sets through Matroids

<table>The lattice of <svg height="15.0875" id="M1447" style="vertical-align:-2.3205pt" version="1.1" viewbox="0 0 70.637497 15.0875" width="70.637497" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path d="M608 643l-12 -19q-56 33 -145 33q-87 0 -154 -42q-68 -42 -68 -115q0 -74 66 -116q67 -42 176 -42q21 0 63 5l29 56q34 66 57 103q43 65 106 113q65 50 122 50q41 0 62 -22t21 -60q0 -67 -87 -144q-88 -76 -217 -110q-46 -101 -119 -186q-72 -83 -136 -108
q71 -23 103 -23q141 0 265 139l19 -14q-131 -156 -281 -156q-84 0 -164 30q-70 -24 -155 -24q-126 0 -126 61q0 24 30 40.5t78 16.5q42 0 85 -16q18 -6 46 -18q26 -11 42 -16q41 26 99 98q58 70 105 158q-26 -3 -53 -3q-50 0 -96.5 10.5t-88.5 31.5t-67 60t-25 90
q0 78 73 130q74 53 182 53q98 0 165 -44zM890 588q0 44 -49 44q-55 0 -109 -80q-54 -81 -89 -183q46 12 104 45q64 35 103.5 81t39.5 93zM266 35v1q-73 43 -125 43q-67 0 -67 -25q0 -33 87 -33q65 0 105 14z" id="x2112"></path></g><g transform="matrix(.017,-0,0,-.017,16.143,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,22.025,12.138)"><path d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" id="x1D440"></path></g><g transform="matrix(.017,-0,0,-.017,39.381,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,45.262,12.138)"><path d="M449 645l-12 -19q-55 31 -136 31q-52 0 -103 -17t-85.5 -54.5t-34.5 -84.5q0 -60 41.5 -96.5t109.5 -43.5q81 125 216 216q137 92 240 92q46 0 71 -19.5t25 -55.5q0 -60 -67 -123q-66 -63 -177 -105q-109 -42 -225 -42q-69 -100 -69 -199q0 -49 31.5 -79.5t85.5 -30.5
q79 0 141 50q61 49 61 108q0 35 -20 54.5t-51 19.5q-53 0 -103 -53q-47 -52 -55 -131l-22 2q0 91 58 158t139 67q55 0 86.5 -26.5t31.5 -76.5q0 -81 -96 -148q-78 -54 -174 -54q-86 0 -141 46q-56 47 -56 127q0 81 48 171q-72 12 -121 55t-49 115q0 75 71 131q72 57 185 57
q91 0 156 -42zM331 355l1 -1q122 0 244 62q73 37 118.5 85.5t45.5 90.5q0 40 -50 40q-73 0 -174 -79q-102 -80 -185 -198z" id="x1D49E"></path></g><g transform="matrix(.017,-0,0,-.017,58.81,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g><g transform="matrix(.017,-0,0,-.017,64.692,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg> (resp., <svg height="16.275" id="M1448" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 88.150002 16.275" width="88.150002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.137)"><path d="M608 643l-12 -19q-56 33 -145 33q-87 0 -154 -42q-68 -42 -68 -115q0 -74 66 -116q67 -42 176 -42q21 0 63 5l29 56q34 66 57 103q43 65 106 113q65 50 122 50q41 0 62 -22t21 -60q0 -67 -87 -144q-88 -76 -217 -110q-46 -101 -119 -186q-72 -83 -136 -108
q71 -23 103 -23q141 0 265 139l19 -14q-131 -156 -281 -156q-84 0 -164 30q-70 -24 -155 -24q-126 0 -126 61q0 24 30 40.5t78 16.5q42 0 85 -16q18 -6 46 -18q26 -11 42 -16q41 26 99 98q58 70 105 158q-26 -3 -53 -3q-50 0 -96.5 10.5t-88.5 31.5t-67 60t-25 90
q0 78 73 130q74 53 182 53q98 0 165 -44zM890 588q0 44 -49 44q-55 0 -109 -80q-54 -81 -89 -183q46 12 104 45q64 35 103.5 81t39.5 93zM266 35v1q-73 43 -125 43q-67 0 -67 -25q0 -33 87 -33q65 0 105 14z" id="x2112"></path></g>
<g transform="matrix(.012,-0,0,-.012,16.137,16.212)"><path d="M655 0h-263v28q46 5 55.5 14.5t-5.5 34.5l-132 214q-69 -104 -122 -202q-19 -32 -10.5 -43.5t58.5 -17.5v-28h-229v28q44 5 70 17t47 40q47 59 164 239l-134 214q-33 50 -54 64t-74 20v28h246v-28q-44 -5 -52 -14t5 -28q32 -55 114 -182q66 102 99 162q19 33 10.5 44.5
t-57.5 17.5v28h228v-28q-56 -6 -80 -19.5t-58 -57.5q-45 -62 -121 -179l157 -246q35 -53 60.5 -70t77.5 -22v-28z" id="x58"></path></g><g transform="matrix(.012,-0,0,-.012,23.919,16.212)"><path d="M729 650v-28q-62 -5 -76 -19.5t-14 -75.5v-404q0 -62 13.5 -76t75.5 -19v-28h-268v28q65 5 79.5 19t14.5 76v199h-341v-199q0 -62 14 -76t76 -19v-28h-263v28q61 5 74.5 19t13.5 76v404q0 62 -14 76t-76 19v28h259v-28q-58 -5 -71 -19t-13 -76v-162h341v162
q0 61 -13.5 75.5t-72.5 19.5v28h261z" id="x48"></path></g>
<g transform="matrix(.017,-0,0,-.017,33.662,12.137)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,39.544,12.137)"><path d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" id="x1D440"></path></g><g transform="matrix(.017,-0,0,-.017,56.9,12.137)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,62.781,12.137)"><path d="M449 645l-12 -19q-55 31 -136 31q-52 0 -103 -17t-85.5 -54.5t-34.5 -84.5q0 -60 41.5 -96.5t109.5 -43.5q81 125 216 216q137 92 240 92q46 0 71 -19.5t25 -55.5q0 -60 -67 -123q-66 -63 -177 -105q-109 -42 -225 -42q-69 -100 -69 -199q0 -49 31.5 -79.5t85.5 -30.5
q79 0 141 50q61 49 61 108q0 35 -20 54.5t-51 19.5q-53 0 -103 -53q-47 -52 -55 -131l-22 2q0 91 58 158t139 67q55 0 86.5 -26.5t31.5 -76.5q0 -81 -96 -148q-78 -54 -174 -54q-86 0 -141 46q-56 47 -56 127q0 81 48 171q-72 12 -121 55t-49 115q0 75 71 131q72 57 185 57
q91 0 156 -42zM331 355l1 -1q122 0 244 62q73 37 118.5 85.5t45.5 90.5q0 40 -50 40q-73 0 -174 -79q-102 -80 -185 -198z" id="x1D49E"></path></g><g transform="matrix(.017,-0,0,-.017,76.329,12.137)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g><g transform="matrix(.017,-0,0,-.017,82.211,12.137)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg>) and <svg height="16.450001" id="M1449" style="vertical-align:-3.41936pt" version="1.1" viewbox="0 0 85.875 16.450001" width="85.875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path d="M608 643l-12 -19q-56 33 -145 33q-87 0 -154 -42q-68 -42 -68 -115q0 -74 66 -116q67 -42 176 -42q21 0 63 5l29 56q34 66 57 103q43 65 106 113q65 50 122 50q41 0 62 -22t21 -60q0 -67 -87 -144q-88 -76 -217 -110q-46 -101 -119 -186q-72 -83 -136 -108
q71 -23 103 -23q141 0 265 139l19 -14q-131 -156 -281 -156q-84 0 -164 30q-70 -24 -155 -24q-126 0 -126 61q0 24 30 40.5t78 16.5q42 0 85 -16q18 -6 46 -18q26 -11 42 -16q41 26 99 98q58 70 105 158q-26 -3 -53 -3q-50 0 -96.5 10.5t-88.5 31.5t-67 60t-25 90
q0 78 73 130q74 53 182 53q98 0 165 -44zM890 588q0 44 -49 44q-55 0 -109 -80q-54 -81 -89 -183q46 12 104 45q64 35 103.5 81t39.5 93zM266 35v1q-73 43 -125 43q-67 0 -67 -25q0 -33 87 -33q65 0 105 14z" id="x2112"></path></g>
<g transform="matrix(.012,-0,0,-.012,16.137,16.213)"><path d="M409 504l-29 -5q-16 60 -44.5 96t-86.5 36q-54 0 -82.5 -32t-28.5 -77q0 -51 30.5 -82.5t96.5 -65.5l35.5 -18t33 -19.5t33.5 -23.5l27 -25.5t24.5 -32t13.5 -36.5t6 -45q0 -80 -62 -134.5t-160 -54.5q-47 0 -98 15q-22 7 -50 21q-8 23 -27 155l30 7q7 -27 18 -52
t30 -51.5t49 -42.5t67 -16q56 0 88 32.5t32 87.5q0 51 -31.5 82t-98.5 67q-80 44 -110 73q-55 53 -55 124q0 75 56 126.5t150 51.5q53 0 126 -23z" id="x53"></path></g><g transform="matrix(.012,-0,0,-.012,21.647,16.213)"><path d="M729 650v-28q-62 -5 -76 -19.5t-14 -75.5v-404q0 -62 13.5 -76t75.5 -19v-28h-268v28q65 5 79.5 19t14.5 76v199h-341v-199q0 -62 14 -76t76 -19v-28h-263v28q61 5 74.5 19t13.5 76v404q0 62 -14 76t-76 19v28h259v-28q-58 -5 -71 -19t-13 -76v-162h341v162
q0 61 -13.5 75.5t-72.5 19.5v28h261z" id="x48"></path></g>
<g transform="matrix(.017,-0,0,-.017,31.375,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,37.257,12.138)"><path d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" id="x1D440"></path></g><g transform="matrix(.017,-0,0,-.017,54.612,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,60.494,12.138)"><path d="M449 645l-12 -19q-55 31 -136 31q-52 0 -103 -17t-85.5 -54.5t-34.5 -84.5q0 -60 41.5 -96.5t109.5 -43.5q81 125 216 216q137 92 240 92q46 0 71 -19.5t25 -55.5q0 -60 -67 -123q-66 -63 -177 -105q-109 -42 -225 -42q-69 -100 -69 -199q0 -49 31.5 -79.5t85.5 -30.5
q79 0 141 50q61 49 61 108q0 35 -20 54.5t-51 19.5q-53 0 -103 -53q-47 -52 -55 -131l-22 2q0 91 58 158t139 67q55 0 86.5 -26.5t31.5 -76.5q0 -81 -96 -148q-78 -54 -174 -54q-86 0 -141 46q-56 47 -56 127q0 81 48 171q-72 12 -121 55t-49 115q0 75 71 131q72 57 185 57
q91 0 156 -42zM331 355l1 -1q122 0 244 62q73 37 118.5 85.5t45.5 90.5q0 40 -50 40q-73 0 -174 -79q-102 -80 -185 -198z" id="x1D49E"></path></g><g transform="matrix(.017,-0,0,-.017,74.042,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g><g transform="matrix(.017,-0,0,-.017,79.923,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg>.</table>

Journal of Applied Mathematics

fig1

Figure 1

Figure 1: Geometric Lattice Structure of Covering-Based Rough Sets through Matroids 