What is the legal term for the implied promise of continued employment?

For example, if you take a new job, and in doing so you have to turn down other jobs, incur moving expenses, etc. You have an expectation that your new employer used due diligence in creating the position and will not end your job for trivial reasons. What is the legal term for this concept?

Whether or not this implied promise actually exists, I know there is a legal term for it, which I've forgotten. That's what I'm asking for.

A contract?

no such thing. if you don't have a contract then it doesn't mean squat.

writing is all the matters

Well, if you and the employer had an agreement in which it was understood that you were to relocate and after doing so you would start working for them, then that is a bilateral contract, even though it is not in writing it is just as binding. If this contract was completed, you moved and then starting working for them, the contract is fulfilled and unless you and the employer had another one, there is no other binding agreement. They can terminate your employment,
"not ending you job for trivial reasons" is in no way a part of the oral agreement you had with the employer. From the information you have provided I think that you are out of luck.

Verbal employer contracts are not legally recognized in a court of law.

No employer is legally obligated to keep you as an employee...that is why they have a probationary period.

That is why you can't find the term.

The term is "expectation of continued employment", which you have a reasonable right to expect if you have been "enticed" away from another job, locale, etc., by an employer who then eliminates the position or terminates you unexpectedly. You may have a legitimate legal claim if this has happened to you, but employment laws vary widely by state...consult an employment attorney in your local area. Good luck.

It's called an "Employment Contract/Agreement" in which all the terms and conditions of employment, including terms of termination are clearly defined. Apart from the issue of whether or not you may pose as a future possible competitor - that particular issue varies according to the legal forum which may restrict that covenant.

Good luck!