Dovgoshey, O.; Luukkainen, J.
(2020)
Let X, Y be sets and let similar to, similar to be mappings with the domains X2 and Y 2 respectively. We say that similar to is combinatorially similar to similar to if there are bijections f : similar to(X2) similar to similar to(Y 2) and g : Y similar to X such that similar to (x, y) = f(similar to(g(x), g(y))) for all x, y similar to Y. It is shown that the semigroups of binary relations generated by sets {similar to-1 (a): a similar to similar to(X 2)} and {similar to-1 (b): b similar to similar to(Y 2)} are isomorphic for combinatorially similar similar to and similar to. The necessary and sufficient conditions under which a given mapping is combinatorially similar to a pseudometric, or strongly rigid pseudometric, or discrete pseudometric are found. The algebraic structure of semigroups generated by {d-1 (r): r similar to d(X 2)} is completely described for nondiscrete, strongly rigid pseudometrics and, also, for discrete pseudometrics d: X2 similar to R.