In some articles I see the word Unix written as "Unix", while in other articles I see the word Unix written as "UNIX".
Does "Unix" and "UNIX" represent the same thing?
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UNIX (in upper case) is a trade mark, currently owned by the Open Group, which is a group of companies and organizations like NASA, the US DOD, IBM, HP, and others (not all American).
On the other hand, like some other trademarks (e.g. "Hoover") the word "unix" or "Unix" is often used as a generic name for "computer software similar to UNIX".
Some writers use "*nix" or "*NIX," presumably to avoid a possible legal challenge over trademark violation, however unlikely that would be.
It's worth noting that companies which market (or used to market) their own variants of UNIX identify them by different names - e.g. HP-UX (Hewlett-Packard), AIX (IBM), IRIX (Silicon Graphics), UNICOS (Cray Research), etc.
Unix and UNIX are different stylizations of the same group of operating systems. It was originally styled as UNIX, and many variations use UNIX as the trademark. Using either styles for all Unix operating systems in general is correct, but you can not claim that a specific version of Unix is named UNIX unless the current or past trademark holders (currently The Open Group) have also called that version a UNIX.
The Open Group will extend the UNIX trademark to any system that qualifies for (and pays to be tested under) the Single UNIX Specification.
A Unix operating system is an operating system that derives from the original UNIX developed at AT&T's Bell Labs.
There have been several forks, some of which have remained proprietary, and some which have become open source.
The Posix specifications was created to describe various components of Unix operating systems, and to ensure interoperability of various Unix and similar systems.
Linux is not a Unix, because it does not derive its source from any Unix distribution, despite being very similar in practice and in feel to popular open Unix distributions of the time. Distros of Linux, however, are overwhelmingly likely to be Posix systems.
One early fork of the original Unix was Research Unix, which AT&T sold both source rights and licensing rights to various universities, to allow their researchers to adapt and redistribute those modification among others computer science academics. The Berkeley Standard Distribution (BSD Unix) came from this fork, and was very influential in other academic versions of Unix. The University of California at Berkeley distributed this version under a very permissive license, effectively granting full open source status to that branch of Unix.
Various child projects of this BSD Unix include FreeBSD, NetBSD, OpenBSD, Mac OS X (now simply just macOS), and iOS. Projects that derive from BSD Unix and BSD licensed software are not required to redistribute their software under similar open source licenses, unlike competing open source licenses such as the GPL.
Other branches of Unix have remained proprietary, and starting with the System V UNIX, it has continued to be released by various copyright holders throughout the years. Proprietary versions of Unix will often seek the Single Unix Specification certification, thus earning the right to use the UNIX trademark. Various UNIX systems include AIX, FTC, EurlerOS, HP-UX, IRIX, K-UX, macOS, OpenServer, Solaris, Tru64 UNIX, UnixWare, and z/OS USS.
Interestingly, the Single UNIX Specification does not specify that a compliant operating system must be derived from a Unix distribution. Thus, a Linux distro can apply for the UNIX trademark.