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We show that some new solutions for a certain system of equations lead to colored Yang\u2013Baxter operators, which are related to an Euler formula for matrices, and the set-theoretical solutions to the Yang\u2013Baxter equation are related to means. A new coalgebra is obtained and studied.<\/jats:p>","DOI":"10.3390\/axioms10030146","type":"journal-article","created":{"date-parts":[[2021,7,2]],"date-time":"2021-07-02T10:06:34Z","timestamp":1625220394000},"page":"146","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On the Colored and the Set-Theoretical Yang\u2013Baxter Equations"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6254-9291","authenticated-orcid":false,"given":"Laszlo Barna","family":"Iantovics","sequence":"first","affiliation":[{"name":"George Emil Palade University of Medicine, Pharmacy, Science and Technology of Targu Mures, Gh. Marinescu 38, 540139 Tg. 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