Abstract
The concept of an involution in the category of rings is extended to universal algebras and is further generalized in that setting. This approach yields four distinct types of involutions on algebras with two binary operations and two distinct types on the category of rings. Subdirectly irreducible universal algebras with involution are considered in detail. A subdirectly irreducible universal algebra with involution is either subdirectly irreducible as an algebra without involution or it is the subdirect product of two subdirectly irreducible algebras and the involution is the exchange involution. An example from the category of rings is given to illustrate that this result is sharp: no direct sum decomposition can be achieved in general. Focus then turns to algebras with two binary operations, particularly near-rings and rings. Subdirectly irreducible objects in the categories of distributive near-rings and of rings are characterized in greater detail, with close attention given to their additive structure.
The third author gratefully acknowledges the kind hospitality he enjoyed at the University of Southwestern Louisiana, Lafayette, and the financial support of the University of Southwestern Louisians and the Hungarian National Foundation for Scientific Research Grant #T016432
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Heatherly, H.E., Lee, E.K.S., Wiegandt, R. (1997). Involutions on Universal Algebras. In: Saad, G., Thomsen, M.J. (eds) Nearrings, Nearfields and K-Loops. Mathematics and Its Applications, vol 426. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1481-0_19
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DOI: https://doi.org/10.1007/978-94-009-1481-0_19
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