TY - JOUR AU - Leeratanavalee, Sorasak AU - Daengsaen, Jukkrit PY - 2022/05/01 Y2 - 2024/03/29 TI - Green’s Relations on Regular Elements of Semigroup of Relational Hypersubstitutions for Algebraic Systems of Type ((m), (n)) JF - Tamkang Journal of Mathematics JA - Tamkang J. Math. VL - 53 IS - 2 SE - Papers DO - 10.5556/j.tkjm.53.2022.3436 UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/3436 SP - 127-146 AB - <div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">Any relational hypersubstitution for algebraic systems of type $</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">(</span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">τ,τ</span><span style="font-size: 7.000000pt; font-family: 'CMSY7'; vertical-align: 4.000000pt;">′</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">) = ((</span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">m</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">i</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">)</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">i</span><span style="font-size: 7.000000pt; font-family: 'CMSY7'; vertical-align: -1.000000pt;">∈</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">I</span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">,</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">(</span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">n</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">j</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">)</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">j</span><span style="font-size: 7.000000pt; font-family: 'CMSY7'; vertical-align: -1.000000pt;">∈</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">J</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">)$ </span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">is a mapping which maps any </span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">m</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">i</span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">-ary operation symbol to an </span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">m</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">i</span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">-ary term and maps any </span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">n</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">j </span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">- ary relational symbol to an </span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">n</span><span style="font-size: 7.000000pt; font-family: 'CMMI7'; vertical-align: -1.000000pt;">j</span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">-ary relational term preserving arities, where </span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">I,J </span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">are indexed sets. Some algebraic properties of the monoid of all relational hypersubstitutions for algebraic systems of a special type, especially the characterization of its order and the set of all regular elements, were first studied by Phusanga and Koppitz [</span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H'; color: rgb(100.000000%, 0.000000%, 0.000000%);">13</span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">] in 2018. In this paper, we study the Green’s relations on the regular part of this monoid of a particular type </span><span style="font-size: 10.000000pt; font-family: 'CMR10';">(</span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">τ,τ</span><span style="font-size: 7.000000pt; font-family: 'CMSY7'; vertical-align: 4.000000pt;">′</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">) = ((</span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">m</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">)</span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">,</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">(</span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">n</span><span style="font-size: 10.000000pt; font-family: 'CMR10';">))</span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">, where </span><span style="font-size: 10.000000pt; font-family: 'CMMI10';">m, n </span><span style="font-size: 10.000000pt; font-family: 'CMSY10';">≥ </span><span style="font-size: 10.000000pt; font-family: 'CMR10';">2</span><span style="font-size: 10.000000pt; font-family: 'MinionPro-Regular-Identity-H';">. </span></p></div></div></div> ER -