Varieties of Boolean inverse semigroups

TL;DR

This paper characterizes the structure and classification of varieties of Boolean inverse semigroups, called biases, linking them to varieties of groups and providing decidability and structural results.

Contribution

It fully describes the varieties of biases in terms of group varieties, including their residual finiteness, the structure of proper varieties, and a correspondence with sequences of group varieties.

Findings

Every free bias is residually finite.

The word problem for free biases is decidable.

Proper varieties of biases contain a largest finite symmetric inverse semigroup.

Abstract

In an earlier work, the author observed that Boolean inverse semi-groups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups: (1) Every free bias is residually finite. In particular, the word problem for free biases is decidable. (2) Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are generalized rook matrices over groups with zero. (3) There is an order-preserving, one-to-one correspondence between proper varieties of biases and certain finite sequences of varieties of groups, descending in a strong sense defined in terms of wreath products by finite symmetric groups.