FOR MEN SWEAR BY ONE GREATER THEN THEMSELVES : anthropoi gar kata tou meizonos omnuousin (3PPAI): (Heb 6:13. Ge 14:22. 21:23, 24. 24:3. 26:20, 26, 28, 31. Mt 23:20-22)

Swear (3660) (Omnuo) means to affirm the truth of a statement by calling on a divine being to execute sanctions against a person if the statement in question is not true (in the case of a deity taking an oath, his divine being is regarded as validating the statement). In this case God's Own Divine being is regarded as validating the statement.

Omnuo is repeated in this middle section of the epistle of Hebrews...

Hebrews 3:11 (note) As I swore in My wrath, 'They shall not enter My rest.'"

Hebrews 3:18 (note) And to whom did He swear that they should not enter His rest, but to those who were disobedient?

Hebrews 4:3 (note) For we who have believed enter that rest, just as He has said, "As I swore in My wrath, They shall not enter My rest," although His works were finished from the foundation of the world.

Hebrews 6:13 (note) For when God made the promise to Abraham, since He could swear by no one greater, He swore by Himself,

Hebrews 6:16 (note) For men swear by one greater than themselves, and with them an oath given as confirmation is an end of every dispute.

Hebrews 7:21 (note) (for they indeed became priests without an oath, but He with an oath through the One who said to Him, "The Lord has sworn And will not change His mind, 'Thou art a priest forever' ")

Omnuo is used in the Septuagint (LXX) in reference to God swearing to keep His covenant to bring Israel into the land (Dt 1:8, 35, 2:14, 4:21, 31, 6:10, 18, 23, 7:8, 12, 13, 8:1, 18, etc > 30x in Deut.)

A recognition of the numerous forensic meanings in Heb 6:16-18, adds considerably to the force of the encouragement extended to the congregation.

The confident hope which God’s oath warrants is justified by the fact that even a human oath puts an end to dispute.

This is a glorious truth. When men are involved in strife and they want to make peace and guarantee the peace, they do it by doing two things. They keep the peace and then they swear or take an oath that they will fulfill their promise. And note: they always swear by someone or something greater than themselves. When they make such an oath, it settles the dispute. They do what they say.

AND WITH THEM AN OATH GIVEN AS CONFIRMATION IS AN END OF EVERY DISPUTE: kai pases autois antilogias peras eis bebaiosin ho horkos kai pases autois antilogias peras: (Dt 32:42. Ge 21:30, 31. Ge 31:53. Ex 22:11. Josh 9:15-20. 2Sa 21:2. Ezek 17:16-20) (Dt 10:20. Ne 5:12. Ps 15:1, 4. Eccl 8:2. Mt 26:63, 64. 2Co 1:23. Phil 1:7. Ex 9:6. Heb 7:7. 12:3. Jude 11)

An oath - The definite article (ho = the) is present so more accurately it reads "the oath".  The invoking of the Lord’s name in the oath meant that one was bound under obligation before God to fulfill that word.

Oath (3727) (horkos from herkos = a fence, an enclosure, that which restrains a person) (See more in depth dictionary discussion of Oath) in simple terms is a solemn statement or claim used to validate a promise. In other words an oath represents  a solemn attestation of the truth or inviolability of one’s words. An oath is a solemn pledge to affirm something said as absolutely true. An oath is a definitive and binding confirmation of the spoken word and invalidates any contradiction of the statement made.

In the OT it was prescribed that oaths should be taken in Yahweh’s name (Deuteronomy 6:13; 10:20), and lying under oath was condemned as a violation of the Third Commandment (Exodus 20:7; Deuteronomy 5:11; Zechariah 5:3,4)

Oath refers to "the confirmation of a compact among men, guaranteeing the discharge of liabilities; in their disputes “the oath is final for confirmation.” This is referred to in order to illustrate the greater subject of God’s “oath” to Abraham, confirming His promise." (Vine)

Noah Webster's 1828 dictionary defines oath as...

A solemn affirmation or declaration, made with an appeal to God for the truth of what is affirmed. The appeal to God in an oath, implies that the person imprecates his vengeance and renounces his favor if the declaration is false, or if the declaration is a promise, the person invokes the vengeance of God if he should fail to fulfill it. A false oath is called perjury.

The formula eis bebaiosin = one that persisted for centuries as a technical expression for a legal guarantee in a transaction. The idea is to cause something to be known as certain, to prove to be true and certain = confirmation, verification.

One might translate "a legal guarantee to every lawsuit".

Vine comments that...

that God confirmed His promise by an oath is only what is done in human custom; for men appeal to God, when taking an oath, as a witness of their sincerity. Further, since a human oath puts an end to debate, God’s oath can remove all doubt in the heart as to the fulfillment of what He has promised. (Vine, W. Collected writings of W. E. Vine. Nashville: Thomas Nelson or Logos)

Guzik has a comment regarding practical application of these truths...

During this time of patient endurance, many Christians get attacked. They wonder if they too will obtain the promise. They often wonder “Will God really come through?” After he had patiently endured, he obtained the promise: God came through for Abraham, even sealing His promise with an oath. In fact, because He could swear by no one greater, He swore by Himself. This oath showed that God’s promises (like His character) are unchanging.

Dispute (485) (Peras)  was a technical term for a legal guarantee = confirmation, legal guarantee

The God of Abraham Praise

1The God of Abraham praise,
Who reigns enthroned above;
Ancient of everlasting days,
And God of love.
Jehovah, great I AM,
By earth and heav’n confessed;
I bow and bless the sacred Name,
Forever blest.

2The God of Abraham praise,
At whose supreme command
From earth I rise, and seek the joys
At His right hand.
I all on earth forsake,
It s wisdom, fame, and pow’r;
And Him my only portion make,
My shield and tow’r.

3He by Himself hath sworn,
I on His OATH depend;
I shall, on eagles’ wings upborne,
To heavn’ ascend;
I shall behold His face,
I shall His pow’r adore,
And sing the wonders of His grace
Forever more.

4The whole triumphant host
Give thanks to God on high
“Hail, Father, Son, and Holy Ghost!”
They ever cry.
Hail, Abraham’s God, and mine!
I join the heav’nly lays;
All might and majesty are Thine,
And endless praise.

IN THE SAME WAY GOD DESIRING EVEN MORE TO SHOW THE HEIRS OF THE PROMISE: en o perissoteron boulomenos (PMPMSN) o theos  epideixai (AAN) tois kleronomois tes epaggelias (singular): (1 Co 12:11. James 1:18. 2 Pe 3:9) ( Ps 36:8. Song 5:1. Isaiah 55:7. Jn 10:10. 1Pe 1:3) (Heb 6:12. Heb 11:7, 9. Ro 8:17. Gal 3:29. Ep 1:11. Ja 2:5. 1Pe 3:7) (Ge 17:7. Ac 2:39. Gal 3:17, 29)

Show (1925) (epideiknumi from epí = upon + deíknumi = show, make known character or significance of something by visual, auditory, gestural, or linguistic means) means to cause to be seen and figuratively meant to prove to be true beyond a doubt and so to demonstrate convincingly.

The writer continues the forensic or legal terminology in this section. Show could be translated “prove” as it is elsewhere, when in Acts 18:28 we read that Apollos proved “in public debate” that Jesus was the Christ. A related verb (apodeiknumi) is used in Acts 25:7 where we read that the Jews were unable to “prove” their charges against Paul in court.

God desired to give proof and demonstrate how committed He was to carry out His promise to the readers (heirs) by giving His oath as surety or guarantee. This is a powerful picture. God Who Himself alone is Judge of all, condescends in a sense to say "I swear to fulfill My promise to you"! This is an incredible truth for finite, created beings to receive from their Creator Who justifiably could have annihilated us all forever! Oh, how deep the canyons of God's mercy to lost sinners such as me.

Inheritance terminology is frequent in Hebrews (see notes Hebrews 1:2, 1:4, 1:14; 6:12, 6:17; 9:15; 11:7, 11:8; 12:17, see Galatians 3:29, see note Romans 8:17)

Promise (1860) (epaggelia from epí = intensifies verbal meaning + aggéllo = to tell, declare) originally referred to an announcement or declaration (especially of a favorable message) but in later Greek came to mean a declaration to do something with the implication of obligation to carry out what is stated (thus a promise or pledge). Epaggelia was primarily a legal term denoting summons, a promise to do or give something, but in the NT speaks primarily of the promises of God.

Epaggelia is used in Hebrews 14 times in 13 verses (27.4% of all 51 NT uses) (See notes Hebrews 4:1, Hebrews 6:12, 6:15, 6:17, Hebrews 7:6, Hebrews 8:6, Hebrews 9:15, Hebrews 10:36, Hebrews 11:9, 11:13, 11:17, 11:33, 11:39)

TDNT summarizes this word group writing that it has the following nuances...

a. The first sense is “to indicate,” “declare,” “declaration,” “report.” b. When the state declares something, it becomes an “order.” c. In law we find the senses “accusation” and “delivery of a judgment.” d. We then find the senses “to declare an achievement,” “to show one's mastery,” “to profess a subject.” e. Another sense is “to offer,” “to promise,” “to vow.” As regards promises, tension between word and deed is felt, so that promises are often seen as worthless. f. A special type of promise is the “promise of money,” and in this sense the idea of a “subscription” or “donation” arises (state liturgies, gifts to rulers at their accession, priests promising gifts in support of their candidature). g. In the Hellenistic period we also find a sacral use for the “proclamation” of a festival. Among all the instances, only one example has been found for the promise of a deity. (Kittel, G., Friedrich, G., & Bromiley, G. W.  Theological Dictionary of the New Testament. Eerdmans)

THE UNCHANGEABLENESS OF HIS PURPOSE INTERPOSED WITH AN OATH: to ametatheton tes boules autou emesiteusen (3SAAI) hork: (Ge 1:9, 18. Job 23:13, 14. Ps 33:11. 110:4. Pr 19:21. Isaiah 14:24, 26, 27. 25:1. 46:10. 54:9, 10. 55:11. Jer 33:20, 21, 25, 26. Malachi 3:6. Ro 4:13, 16. 11:29. Ja 1:17. of his. Ex 19:6. unchangeable Acts 20:27. Interposed interposed himself.  Heb 6:16. Ge 26:28. Ex 22:11. Gal 3:20. Oath. He 8:6. Jer 33:20, 21) (See Immutability; Unchangeable.) (See God's attribute - Immutable)

Unchangeableness (276) (amethathetos from a = without, + metatíthemi = change condition or place, transfer) literally means that which does not change from one state to another and gives us the truth that God's purpose is immutable and unchangeable. Spoken of God's constitutional will. This Greek word was used in wills and contracts and signified stipulation that could not be annulled. Here the writer conveys the strong certainty of IRREVOCABILITY of God's purpose as expressed in His promise (His Word) and in His oath.

God actively actually intervened as surety by declaring His oath. The picture is God pledging Himself as SURETY or GUARANTEE in regard to His "contracted obligation".