By semigroup, we mean a Hausdorff topological space together with a continuous associative multiplication. We are concerned here with the existence of local subsemigroups, which are arcs, in compact connected semigroups whose principal left ideals are linearly ordered by C. In addition, we seek local cross sections at certain idempotent elements in such semigroups. In [5 ], Koch has shown that if S is a compact connected semigroup with identity, then S contains an arc. In [7], Mostert and Shields obtained local one parameter semigroups at the identity. Hunter, in [2], and Hunter and Rothman, in [3], have obtained arcs which are local subsemigroups and certain local cross sections in compact connected abelian topological semigroups. Our approach is similar to that in [2] and [3], and concerns noncommutative semigroups. We follow the notation and terminology of [1; 2; 3; 11 ]. In particular, a nonempty subset L(R, I) of a semigroup S is a left (right, two sided) ideal of S if SLCL(RSCR, SIUISC-I). The left ideal L is principal if for some xGL, L = { x } UJSx. We denote by Lp the set of those x in S for which { x } UJSx = { p } UJSp. The symbol K is reserved for the minimal ideal of a semigroup (when S is compact, K exists), and E is used to denote the set of idempotent elements of S (when S is compact, E is nonvoid). If eCE, the maximal subgroup of S containing e is designated by He. It is known that the sets Lp form an upper semi-continuous decomposition of S, when S is compact. We let S' be the associated hyperspace of this decomposition and 4 be the natural mapping. That is 4: S-->S' is given by +(x) = { Lx }. We are interested in the cases when S' is again a semigroup and 4) is a homomorphism. In particular, Theorem 1 gives necessary and sufficient conditions that S' be a standard thread [I]. DEFINITION. A semigroup S is said to be left linearly quasi-ordered if for each x and yin S, either {x}JUSxC{y}USy or {y}JUSyC{x} USx. It is easy to see that the order induced on S, x _y if and only if {x } USx C { y } USy, is a continuous quasi-order in the sense of Nachbin [8] and Ward [10]. It follows that each compact subset of S has a maximal element. We assume for the remainder of this note, that S is a compact