Assume that S is a right normal orthodox semigroup with an inverse transversal S[degrees].
Let S is a right normal orthodox semigroup with an inverse transversal S[degrees], Blyth and Almeida Snatos in  proved that there is an order-preserving bijection from the set of all locally maximal S[degrees]-cones to the set of all left amenable orders definable on S and the natural partial order is the smallest left amenable partial order(see theorems 7 and 11 in ).
Suppose that S is a locally inverse semigroup with an inverse transversal S[degrees].
Finally, it is established that for any (but fixed) the inverse transversal S[degrees] of S, there is an order-preserving bijection from the set of all amenable partial orders on S to the set of all amenable partial orders on S[degrees] and so every amenable partial order on S[degrees] is uniquely extended to an amenable partial order on S.
This last part, with proposition 3.9 at its core, is not connected to minimal transversals as such but only substantiates our remark in section 1 that the complemented HG of H = (V, E) need not always turn out to be a HG on V as advanced in .
The strength of 3.2 is that it characterizes minimal transversals in all simple hypergraphs, not being restricted to those in which every edge is a minimal transversal.
Gottlob, 1995, "Identifying the minimal transversals of a hypergraph and related problems", SIAM Journal on Computing, 24, pp 12781034.
T [member of] [2.sup.v*] is called a transversal in H if T [intersection] A [not equal to] [phi] for each A [member of] E which criterion we rephrase as "T meets every member of E." If T does not meet A [member of] E, then we say T bypasses A, and call T a bypasser.