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In this third part relative simplicial monoids are analyzed. Their Moore length is introduced and the equivalence is proven between relative simplicial monoids of Moore length 1, and relative categories of monoids in Part I. This equivalence is obtained in one direction by truncating a simplicial monoid at the first two degrees; and in the other direction by taking the simplicial nerve of a relative category.<\/jats:p>","DOI":"10.1007\/s10485-020-09603-z","type":"journal-article","created":{"date-parts":[[2020,8,20]],"date-time":"2020-08-20T05:22:44Z","timestamp":1597900964000},"page":"1-29","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Crossed Modules of Monoids III. 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