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In this paper, necessary and sufficient conditions of the one-sided reverse order law $$(ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # a # , the two-sided reverse order law $$(ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # a # and $$(ba)^{\tiny {\textcircled {\tiny \#}}}=a^{\tiny {\textcircled {\tiny \#}}}b^{\tiny {\textcircled {\tiny \#}}}$$ ( b a ) # = a # b # for the core inverse are given in rings with involution. In addition, the mixed-type reverse order laws, such as $$(ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}(abb^{\tiny {\textcircled {\tiny \#}}})^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # ( a b b # ) # , $$a^{\tiny {\textcircled {\tiny \#}}}=b(ab)^{\#}$$ a # = b ( a b ) # and $$(ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}$$ ( a b ) # = b # a # , are also considered.
Mediterranean Journal of Mathematics – Springer Journals
Published: Jun 5, 2018
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