Summary: The commuting regularity degree, \(\mathrm{dcr}(S)\) of a non-group semigroup \(S\) is defined and studied recently by the authors [Creat. Math. Inform. 24, No. 1, 43–47 (2015; Zbl 1349.20044)], where \(\mathrm{dcr}(S)\) is the probability that a pair \((x,y)\) of the elements of \(S\) is a commuting regular pair (the pair \((x,y)\) is called a commuting regular pair if for some element \(z\in S\), \(xy=(yx)z(yx)\)). This definition is an identifier in characterization of the commuting regular semigroups specially when they are non-group. When \(\mathrm{dcr}(S)=1\), then \(S\) is called a commuting regular semigroup. Among all of the recent studies on non-group non-commutative semigroups, \(\mathrm{dcr}(S)\) is less than or equal \(\frac12\). A natural question is finding finite non-group non-commutative semigroups for which \(dcr\) achieves the maximum value. In this paper, we prove that for a non-group regular quasi-commutative semigroup \(S\), \(\mathrm{dcr}(S)=1\). Moreover, the converse does not hold. Indeed, by considering the infinite class of finitely presented semigroups \[ S(n)=\langle a,b\mid a^{2^{n-1}+1}=a,b^2=a^{2^{n-2}},ba=a^{2^{n-1}-1}b\rangle \] of order \(2^n+1\), we show that for every \(n\geq4\), \(\mathrm{dcr}(S(n))=1\), however, \(S(N)\) is non-regular and non quasi-commutative, where a semigroup \(S\) is said to be quasi-commutative if for all \(x,y\in S\) there exists a positive integer \(r\) such that \(xy=y^rx\). Our final results concerning \(\mathrm{dcr}(S)=1\), are about the direct and semidirect products of finite semigroups.