Regular and unit-regular elements in various monoids of transformations

Let T(X) be the full transformation monoid on a nonempty set X. An element f of T(X) is said to be semi-balanced if the collapse of f is equal to the defect of f. In this paper, we prove that an element of T(X) is unit-regular if and only if it is semi-balanced. For a partition P of X, we characterize unit-regular elements in the monoid T(X,P) = {f is an element of T(X)vertical bar(for all X-i is an element of P)(Sigma X-j is an element of P&)X(i)f subset of X-j} under composition. We characterize regular elements in the submonoids sigma(X,P) = {f is an element of T(X,P)|(for all X-i is an element of P)Xf boolean AND X-i not equal null } and omega(X,P) = {f is an element of T(X,P)|(for all x,y is an element of X,x not equal y)(x,y) is an element of E & xf not equal yf} of T(X,P), where E is the equivalence induced by P. We also characterize unit-regular elements in sigma(X,P), omega(X,P), and the other two known submonoids of T(X,P).