One-sided duality theorems

The phenomenon of duality is inherent in all sections of mathematics and underlies many special duality theorems that assert the possibility of dual transitions – transfers of mathematical statements from one area of mathematics to another. All known duality theorems are based on properties of special mathematical structures and are bilateral in nature, i. e. they assume dual transitions in one and other directions. The present paper is devoted to a new understanding of dual transitions as transitions from internal (respectively external) descriptions of sets to external (respectively internal) descriptions of sets dual to them. Special attention is paid to one-way dual transitions, one-way duality theorems. The abstract constructions (one-sided duality theory) are based on the notion of dual scheme, which, in turn, is based on the notion of weakened involution – a fully isotone mapping. In this case, any fully isotone mapping has a conditionally inverse mapping which is also fully isotone. The authors distinguish four dual schemes, each of which plays its strictly defined role in matters of external and internal description of sets. Any dual scheme is represented as a set of two diagrams connected by mutually inverse transitions to conditionally inverse mappings.