en تخصصي Special پژوهشي Research paper <div style="text-align: justify;">We prove that continuous sentences preserved by the ultramean construction (a generalization of the ultraproduct construction) are exactly those sentences which are approximated by linear sentences. Continuous sentences preserved by linear elementary equivalence are exactly those sentences which are approximated in the Riesz space generated by linear sentences. Also, characterizations for linear $Delta_n$-sentences and positive linear theories will be given.</div> Continuous logic, Ultramean, Linear formula, $Sigma_n$-formula, Positive formula. 75 86 http://ijmsi.ir/browse.php?a_code=A-10-430-2&slc_lang=en&sid=1 S. M. Bagheri bagheri@modares.ac.ir 10031947532846009582 10031947532846009582 Yes Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran F. Fadai fadai@modares.ac.ir 10031947532846009583 10031947532846009583 No Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran