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Recently, an algorithm has been developed to determine implications <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{M}\\Rightarrow _{\\textrm{lex}_*}\\textrm{N}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>M<\/mml:mtext>\n                    <mml:mmultiscripts>\n                      <mml:mo>\u21d2<\/mml:mo>\n                      <mml:mmultiscripts>\n                        <mml:mtext>lex<\/mml:mtext>\n                        <mml:mrow>\n                          <mml:mrow\/>\n                          <mml:mo>\u2217<\/mml:mo>\n                        <\/mml:mrow>\n                        <mml:mrow\/>\n                      <\/mml:mmultiscripts>\n                      <mml:mrow\/>\n                    <\/mml:mmultiscripts>\n                    <mml:mtext>N<\/mml:mtext>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> between them. We show here that this algorithm reduces to constructing a partial term corresponding to <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{N}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mtext>N<\/mml:mtext>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> from a partial term corresponding to\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{M}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mtext>M<\/mml:mtext>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Moreover, we prove that this is further equivalent to the corresponding implication between the weak versions of these properties, i.e., the one where only strong monomorphisms are considered instead of all monomorphisms.<\/jats:p>","DOI":"10.1007\/s10485-024-09790-z","type":"journal-article","created":{"date-parts":[[2024,10,26]],"date-time":"2024-10-26T17:01:44Z","timestamp":1729962104000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Partial Algebras and Implications of (Weak) Matrix Properties"],"prefix":"10.1007","volume":"32","author":[{"given":"Michael","family":"Hoefnagel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pierre-Alain","family":"Jacqmin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,10,26]]},"reference":[{"key":"9790_CR1","first-page":"175","volume":"29","author":"J Ad\u00e1mek","year":"1988","unstructured":"Ad\u00e1mek, J., Herrlich, H., Rosick\u00fd, J.: Essentially equational categories. 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