Ideals in BIT speciale varieties

TL;DR

This paper characterizes ideals in BIT speciale varieties, showing that finite sets of ideal terms determine all ideals, and compares these sets across various algebraic structures.

Contribution

It introduces finite sets of ideal terms that characterize ideals in finitary BIT speciale varieties, linking these to known structures in specific algebraic varieties.

Findings

Finite sets of ideal terms determine all ideals in finitary BIT speciale varieties.

Specific term sets are identified for groups, rings, and other algebraic structures.

The identified term sets coincide or intersect significantly with previously known sets for these structures.

Abstract

Ideals in BIT speciale varieties are characterized. In particular, it is proved that, for any finitary BIT speciale variety, there is a finite set of ideal terms determining ideals. Several ideal term sets of this kind are given. For the variety of groups one of these sets consists of the terms $y_1y_2$, $xyx^{-1}$, $x^{-1}y^{-1}x$ (and $y$ which can be ignored), while for rings it consists of the terms $y_1+y_2$, $-y$, $xy$, $yx$. For each of the following varieties -- groups with multiple operators, semi-loops, and divisible involutory groupoids -- one of the term sets found in this paper almost coincides with the term sets found earlier for these particular varieties by resp. Higgins, B\v{e}lohl\'{a}vek and Chajda, and Hala\v{s}. The coincidence is precise in the case of divisible involutory groupoids. For loops (resp. loops with operators), the intersection of one of the term sets found in this paper with the term sets found earlier for loops (resp. for loops with operators) by Bruck (resp. by Higgins) forms the major part of both sets.