compleat the words ( o……. k……. ) ?

It is unlikely that the letters O-K represent words. It is more a short form of ‘okay’ There are many, many theories of its origin. You can read about them here:

http://www.miketodd.net/encyc/okay.htm

The historical record shows that O.K. appeared as an abbreviation for “oll korrect” (a conscious misspelling of “all correct”) in Boston newspapers in 1839, and was reinterpreted as “Old Kinderhook” in the 1840 United States presidential election. Because it is a recent word born of word play, and because it is so widely used, O.K. has also invited many folk etymologies. These competing theories are not supported by the historical written record, except in that folk and joke etymologies influenced the true history of the word. Since the 19th century, the word has spread around the world, the okay spelling of it first appearing in British writing in the 1860s. Spelled out in full in the 20th century, ‘okay’ has come to be in everyday use among English speakers, and borrowed by non-English speakers.

The wordplay origin of O.K. invited folk etymology and joke etymology from the beginning. Eventually there appeared folk etymologies that were not connected with either word play or the 1840 Presidential election. In particular, in 1859, a Tennessee historian named Albigence Waldo Putnam misread an appearance of O.R. in a 1790 missive by Andrew Jackson as O.K.. This made Andrew Jackson the dominant theory of the origin of O.K. until it was disproven by Woodford Heflin in 1941 using photographic analysis.

According to Read, an English professor at the University of Alabama named W. S. Wyman attributed OK to the Choctaw word “okeh”, which means “it is so”, in 1885. This theory was endorsed by President Woodrow Wilson. Unlike Read’s own scholarship, this etymology lacks a clear historical record. Nonetheless, this theory remains popular among some students of American Indian culture.[7]

It has been suggested that in World War II the term “zero killed” was used when a unit suffered no casualties in combat, and that this was then shortened to 0K. This proposed etymology is grossly anachronistic, since by this time the term had been widely used for a full century. The same theory has also been applied to the Civil War, but this is also anachronistic.

Another story is that the expression came from a quality control system in some company, in which some inspector with the initials O.K. provided final approval. Some versions of this story include impossibly anachronistic choices for the company such as the Ford Motor Company, as well as implausible employee names such as “Omar Kulemsky”.

The most common theories are that it either comes from “oll korrect,” a misspelling of “all correct” or from “Old Kinderhook,” the nickname of US president Martin Van Buren. http://en.wikipedia.org/wiki/Okay

It canNOT be said to be short for “okay,” which is a casual phonetic spelling of O.K. that came later.

it means an agreement of something, okay

ok is short for OKay