What is the idea behind creation of a compound word - 处决?
'处' in '处决' means "to sentence/ to put to"


The basic meaning of the verb '处' is 'to place', and 'to sentence' is an extended meaning. 


"处以極刑" = "to place someone in the situation of receive capital punishment"= "to sentence someone to capital punishment"



  '决' in '处决' means "to execute"


The basic meaning of the verb '决' is 'to determine' and 'to execute' is an extended meaning. 


to determine-->  to put an end to--> to put to death



  处决 as a compound word is a verb means 1.to decide a matter 2. to execute/ to put to death


Examples:


公司上下事情由他全權處決-  He alone decides matters on all affairs of the company 
處決某人 - Execute someone.