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New partial orders on Rickart *-rings

This paper introduces and investigates new partial orders on Rickart *-rings defined by additional conditions on star and one-sided star partial orders, establishing that their down-sets are order-isomorphic to subsets of self-adjoint idempotents and form lattices in regular rings while providing characterizations for supremum and infimum operations.

Original authors: Cecilia Rossana Cimadamore, Laura Alicia Rueda, Melina Vanina Verdecchia

Published 2026-07-20

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CC BY 4.0

Original authors: Cecilia Rossana Cimadamore, Laura Alicia Rueda, Melina Vanina Verdecchia

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a universe made entirely of numbers and symbols that follow strict rules of interaction, like a massive, invisible game of chess where the pieces can flip themselves inside out. This is the world of abstract algebra, specifically a branch dealing with "rings" and "stars." In this context, a "ring" isn't a piece of jewelry, but a collection of objects that you can add and multiply together. The "star" is a special rule that flips an object over, kind of like looking at a reflection in a mirror. Mathematicians love these structures because they are the hidden blueprints behind everything from computer encryption to quantum physics.

Within this universe, there are special objects called "Rickart *-rings." Think of these as highly organized neighborhoods where every resident (an element) has a unique "ID card" (a self-adjoint idempotent) that tells you exactly where they live and how they interact with their neighbors. For a long time, mathematicians have used a "star order" to compare these residents, asking questions like, "Is this person a smaller version of that one?" or "Do they fit inside each other's shadows?" It's a way of sorting the chaos into a neat hierarchy. But just like in real life, sometimes one rule isn't enough to describe a complex relationship. What if two people are related in a specific way, but also happen to share a secret handshake? That's the puzzle this paper tackles: creating new, more specific ways to sort these mathematical objects by adding extra conditions to the existing rules.

The authors of this paper, working from Argentina, decided to spice up the existing sorting system. They took the standard "star order" and the "one-sided star orders" (which are like checking if one person fits inside another's left or right pocket) and combined them with a new, quirky condition: squaring. In math, squaring a number means multiplying it by itself ( $a \times a$ ). The researchers asked, "What happens if we only compare two objects if, in addition to fitting the standard rules, the first object squared equals the first object times the second?"

They didn't just guess; they built nine brand-new "partial orders." You can think of a partial order as a set of rules for a game of "who is bigger?" In this new game, they created specific categories like the "right-star-left-square" order or the "star-square" order. Each of these is a unique lens through which to view the ring. The paper proves that these new rules actually work—they are consistent, logical, and don't lead to contradictions. They are real, valid ways to organize the mathematical universe.

One of the coolest discoveries in the paper is how these new, complicated relationships can be simplified. The authors found that if you look at all the objects that are "smaller" than a specific big object under these new rules, you can map them perfectly onto a simpler group of "self-adjoint idempotents." Imagine taking a complex, tangled knot of relationships and realizing it's actually just a clean, straight line of simple blocks. This "order-isomorphism" is a powerful tool. It means that instead of wrestling with the messy, complex rules of the new orders, mathematicians can translate the problem into the language of these simple blocks, solve it there, and translate the answer back. It's like realizing that a difficult puzzle is actually just a picture of a cat once you turn it sideways.

The paper also dives into what happens when the ring is "regular." In this math world, "regular" means the ring is well-behaved and predictable. When the ring is regular, the authors proved that these new groups of "smaller" objects form a "lattice." In everyday terms, a lattice is a structure where if you pick any two items, you can always find a unique "smallest common bigger thing" (supremum) and a "biggest common smaller thing" (infimum). It's like having a family tree where you can always find the closest common ancestor or the youngest common descendant. The paper doesn't just say these exist; it gives you the exact recipe to calculate them using the "Moore-Penrose inverse," a special mathematical tool that acts like a reverse button for these objects.

Finally, the researchers extended these findings to an even more robust type of ring called a "Baer *-ring." They showed that even if you have a huge, non-empty crowd of objects, as long as they are all "under" a certain limit, you can still find their collective highest and lowest points. The paper provides the exact formulas for these points, ensuring that no matter how complex the group gets, the structure remains solid and predictable.

In short, this paper is about building better, more specific rulers for measuring mathematical objects. By adding the condition of "squaring" to the existing rules, the authors created a richer, more detailed map of the mathematical landscape. They proved that these new maps are valid, showed how to translate complex problems into simple ones, and demonstrated that in well-behaved rings, these new maps always have a clear "top" and "bottom" for any group of items. It's a work of pure structural beauty, turning a chaotic collection of symbols into a perfectly ordered garden.

*Technical Summary: New Partial Orders on Rickart -Rings

Problem Statement
The paper addresses the structural analysis of partial orders within the context of Rickart *-rings. While the star partial order ( $\ast \preceq$ ) and its one-sided variants (left star $\ell\ast \preceq$ and right star $r\ast \preceq$ ) are well-established in the literature for complex matrices, bounded linear operators, and *-rings, this work identifies a need to explore new binary relations. These new relations are formed by combining existing star orders with additional algebraic conditions involving squares of elements (specifically $a^2 = ab$ , $a^2 = ba$ , or $a^2 = ab = ba$ ). The primary objective is to define these new orders, establish their fundamental properties, and investigate the lattice structure of the down-sets (elements below a given element) induced by these relations, particularly in regular Rickart *-rings and regular Baer *-rings.

Methodology
The authors employ a structural approach rooted in the theory of Rickart *-rings and generalized inverses. The methodology proceeds through the following steps:

Definition and Verification: The authors define nine new binary relations on a Rickart *-ring $A$ by intersecting the standard star orders ( $\ast \preceq, \ell\ast \preceq, r\ast \preceq$ ) with conditions $(\ell\Box)$ , $(r\Box)$ , or $(\Box)$ . They rigorously prove that these relations satisfy the axioms of partial orders (reflexivity, antisymmetry, and transitivity).
Isomorphic Representation: A central methodological tool is the construction of order-isomorphisms. The authors map the down-set of an element $b$ , denoted $[0, b]_\alpha = \{a \in A : a \alpha \preceq b\}$ , to specific subsets of self-adjoint idempotents $E_\alpha^b \subseteq E(A)$ . These subsets are defined by conditions on the idempotents (e.g., $p \leq r_p(b)$ and commutativity or specific product relations involving $b$ ).
Lattice Analysis: Leveraging the known fact that the set of self-adjoint idempotents $E(A)$ in a regular Rickart *-ring forms a lattice, the authors transfer this lattice structure to the down-sets via the established isomorphisms.
Characterization: Using these isomorphisms, the authors derive explicit characterizations of elements $a$ that are below a given $b$ under each new order. They also analyze the existence and explicit formulas for suprema and infima of pairs (and nonempty subsets in Baer *-rings) within these down-sets.

Key Contributions

Introduction of New Partial Orders: The paper formally introduces and validates nine new partial orders on Rickart *-rings, denoted as $\alpha \preceq$ where $\alpha \in \{r\ast|\ell\Box, r\ast|r\Box, r\ast|\Box, \ell\ast|\ell\Box, \ell\ast|r\Box, \ell\ast|\Box, \ast|\ell\Box, \ast|r\Box, \ast|\Box\}$ .
Extension of Known Orders: The authors demonstrate that these new orders generalize existing concepts. Specifically:
- The order $\ell\ast|r\Box \preceq$ restricted to core-invertible elements coincides with the core partial order.
- The order $r\ast|\ell\Box \preceq$ restricted to dual core-invertible elements coincides with the dual core partial order.
- The orders $\ell\ast|\Box \preceq$ , $r\ast|\Box \preceq$ , and $\ast|\Box \preceq$ relate to the sharp partial order and one-sided sharp orders when restricted to group-invertible elements.
Order-Isomorphisms: The paper establishes that for any $b \in A$ , the down-set $[0, b]_\alpha$ is order-isomorphic to a specific subset of self-adjoint idempotents $E_\alpha^b$ . This reduces the study of the order structure of arbitrary ring elements to the study of idempotents, which are more tractable.
Lattice Structure in Regular Rings: The authors prove that if $A$ is a regular Rickart *-ring, then the down-set $[0, b]_\alpha$ is a lattice for every new order $\alpha$ . Furthermore, they provide explicit formulas for the supremum and infimum of any two elements $a_1, a_2 \in [0, b]_\alpha$ in terms of the Moore-Penrose inverses and the lattice operations of the associated idempotents (e.g., $a_1 \vee a_2 = b(r_p(a_1) \vee r_p(a_2))$ ).
*Generalization to Baer -Rings: The results regarding the existence of suprema and infima are extended to regular Baer *-rings for nonempty subsets of elements, provided the subset is bounded above.

Results

Characterization of Elements: For each new order, the paper provides necessary and sufficient conditions for $a \alpha \preceq b$ . For instance, $a r\ast|\ell\Box \preceq b$ if and only if there exists a unique self-adjoint idempotent $p$ such that $a = bp$, $p \leq r_p(b)$ , and $p b_r p = p b_r$ (where $b_r = r_p(b)b$ ).
Lattice Properties: It is proven that in a regular Rickart *-ring, the down-set $[0, b]_\alpha$ is not only a lattice but a sublattice of the larger poset $[0, b]_\beta$ (where $\beta$ is a weaker order, such as the standard star order).
Explicit Formulas: The supremum and infimum of elements $a_1, a_2$ bounded by $b$ are explicitly calculated using the lattice operations of the underlying idempotents. For right-sided orders, the formulas involve $r_p(a_i)$ ; for left-sided orders, they involve $l_p(a_i)$ .
Isomorphism of Intervals: For specific orders ( $\alpha \in \{r\ast|\Box, \ell\ast|\Box, \ast|\Box\}$ ), the interval $[a, b]_\alpha$ is shown to be isomorphic to the down-set $[0, b-a]_\alpha$ .

Significance
The paper claims significance by providing a unified framework that extends several well-known partial orders (core, dual core, sharp) from specific subsets of invertible elements to the entire set of elements in a Rickart *-ring. By establishing order-isomorphisms between these down-sets and subsets of self-adjoint idempotents, the authors simplify the analysis of these complex algebraic structures. The demonstration that these down-sets form lattices in regular rings offers a robust structural understanding of the ordering properties in these rings, facilitating further algebraic investigations. The extension of these results to regular Baer *-rings for arbitrary bounded subsets further broadens the applicability of the findings to more general ring-theoretic contexts.

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