Abstract
We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents.
Highlights
For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits
In this note we propose to generalize the concept of regularity for semigroups in two different aspects simultaneously: (1) higher regularity, which can be informally interpreted that each element has several inverse elements; (2) higher arity, which extends the binary multiplication to that of arbitrary arity, i.e., the consideration of polyadic semigroups
The higher regularity concept was introduced in semisupermanifold theory [8] for generalized transition functions, which gave rise to the development of a new kind of so called regular obstructed category [9] and to their application to Topological Quantum Field Theory [10], the Yang-Baxter equation [11] and statistics with a doubly regular Rmatrix [12]
Summary
The concept of regularity was introduced in [1] and widely used in the construction of regular and inverse semigroups (see, e.g., [2,3,4,5,6,7], refs therein). In this note we propose to generalize the concept of regularity for semigroups in two different aspects simultaneously: (1) higher regularity, which can be informally interpreted that each element has several inverse elements; (2) higher arity, which extends the binary multiplication to that of arbitrary arity, i.e., the consideration of polyadic semigroups. The higher regular idempotents can be introduced, and their commuting leads to the higher inverse semigroups. To get polyadic idempotents (by analogy with the ordinary regularity for semigroups) we introduce the so called sandwich regularity for polyadic semigroups (which differs from [16]). In trying to connect the commutation of idempotents with uniqueness of inverse elements (as in the standard regularity) in order to obtain inverse polyadic semigroups, we observe that this is prevented by the middle elements in the sandwich polyadic regularity conditions. Further investigations are needed to develop the higher regular and inverse polyadic semigroups
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