An extension of a Y. C. Yang theorem

The main purpose of this paper is to extend the theorem of Y. C. Yang stating that lattice-ordered skew fields are totally ordered iff squares are positive (Yang in Am Math Mon 113(3):266-267, 2006). The principal tools are an extension of a result of Teh concerning weak l-groups (Teh in Proc Edinb Math Soc 13(1):123-124, 1962) and a connection with Benado multilattices introduced in Benado (Bul Sti Sect Sti Mat Fiz 5:41-48, 1953); see also Benado (Czechoslov Math 5(3):308-344, 1955), Rudeanu and Vaida (J Mult-Valued Logic Soft Comput 20(3-4):265-307, 2013). Within the proofs below, the properties related to po-partial monoids or to po-semiring-like systems are stated such that to require each time only what is strictly needed for the stated result.