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commonThms={a·bb·a,a(a·b)·(a·a),(a·a)·(a·a)a,(a·b)·(b·b)b,a(a·b)·(a·(b·c)),a(a·b)·(a·(c·b)),a(a·b)·((b·c)·a),a(a·b)·((c·b)·a),a(b·a)·(a·(b·c)),a(b·a)·((b·c)·a),a(b·(b·b))·(a·a),a((a·b)·a)·(a·a),a·aa·(a·(a·a)),a·aa·((a·a)·a),a·ba·((a·b)·a),a·bb·(b·(a·a)),a·b((a·b)·a)·a,a·(a·a)b·(b·b),a·(a·a)(b·b)·b,a·(a·(b·a))b·a,a·(b·a)a·(b·b),a·(b·a)(b·a)·a,a·(b·(a·b))a·a,a·(b·(b·b))a·a,a·(a·a)b·(b·b),a·(a·a)c·(c·c),a·(a·a)(b·b)·b,a·(a·a)(c·c)·c,a·(a·b)a·(b·b),a·(a·b)(b·b)·a,a·(a·c)a·(c·c),a·(a·c)(c·c)·a,a·(b·a)a·(b·b),a·(b·b)a·(a·b),a·(b·b)a·(b·a),a·(c·a)a·(c·c),a·(c·c)a·(a·c),a·(c·c)a·(c·a),b·(a·a)b·(a·b),b·(a·a)b·(b·a),b·(a·b)b·(a·a),b·(a·b)(a·a)·b,b·(b·a)b·(a·a),b·(b·a)(a·a)·b,b·(b·b)a·(a·a),b·(b·b)c·(c·c),b·(b·b)(c·c)·c,b·(b·c)b·(c·c),b·(b·c)(c·c)·b,b·(c·b)b·(c·c)};
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ResourceFunction["FindFiniteModels"][#,2]&/@commonThms;
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Length/@%
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{8,6,8,4,4,4,6,6,2,4,4,5,16,13,10,6,10,6,4,10,12,9,10,8,6,6,4,4,10,6,10,6,12,10,12,12,10,12,12,10,12,7,10,6,6,6,4,10,6,12}
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ResourceFunction["FindFiniteModels"][#,3]&/@commonThms;
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Length/@%
Out[]=
{729,188,2916,216,10,10,32,32,0,10,67,450,12167,7148,1113,44,1208,1620,492,1208,1891,1258,760,1014,1620,1620,492,492,973,119,973,119,1891,973,1891,1891,973,1891,1891,973,1891,219,973,119,1620,1620,492,973,119,1891}
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Position[%10,0]
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{{9}}
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commonThms[[9]]
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a(b·a)·(a·(b·c))
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AxiomaticTheory["WolframAxioms"]
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((a.·b.)·c.)·(a.·((a.·c.)·a.))c.
∀
{a.,b.,c.}
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ResourceFunction["FindFiniteModels"][a(b·a)·(a·(b·c)),4]
Out[]=
{95356335}CenterDot{{0,0,1,1},{2,2,3,3},{0,0,1,1},{2,2,3,3}},{100334085}CenterDot{{0,0,1,1},{3,3,2,2},{3,3,2,2},{0,0,1,1}},{572684151}CenterDot{{0,2,0,2},{0,2,0,2},{1,3,1,3},{1,3,1,3}},{584965410}CenterDot{{0,2,0,2},{3,1,3,1},{3,1,3,1},{0,2,0,2}},{1010592105}CenterDot{{0,3,3,0},{0,3,3,0},{1,2,2,1},{1,2,2,1}},{1016478870}CenterDot{{0,3,3,0},{2,1,1,2},{0,3,3,0},{2,1,1,2}},{1308675202}CenterDot{{1,0,3,2},{0,0,0,0},{3,0,3,0},{2,0,0,2}},{1431207270}CenterDot{{1,1,1,1},{1,0,3,2},{1,3,3,1},{1,2,1,2}},{1719815850}CenterDot{{1,2,1,2},{2,0,0,2},{1,0,3,2},{2,2,2,2}},{1774414275}CenterDot{{1,2,2,1},{3,0,0,3},{1,2,2,1},{3,0,0,3}},{2005436535}CenterDot{{1,3,1,3},{2,0,2,0},{2,0,2,0},{1,3,1,3}},{2110586702}CenterDot{{1,3,3,1},{3,0,3,0},{3,3,3,3},{1,0,3,2}},{2526462915}CenterDot{{2,1,1,2},{2,1,1,2},{3,0,0,3},{3,0,0,3}},{2779857237}CenterDot{{2,2,1,1},{2,3,0,1},{1,0,0,1},{1,1,1,1}},{2864624037}CenterDot{{2,2,2,2},{2,3,3,2},{2,3,0,1},{2,2,1,1}},{2941276335}CenterDot{{2,2,3,3},{1,1,0,0},{1,1,0,0},{2,2,3,3}},{2985295937}CenterDot{{2,3,0,1},{3,3,0,0},{0,0,0,0},{1,0,0,1}},{3204444337}CenterDot{{2,3,3,2},{3,3,3,3},{3,3,0,0},{2,3,0,1}},{3722283144}CenterDot{{3,1,3,1},{3,1,3,1},{2,0,2,0},{2,0,2,0}},{3835708416}CenterDot{{3,2,1,0},{2,2,0,0},{1,0,1,0},{0,0,0,0}},{4004177056}CenterDot{{3,2,3,2},{2,2,2,2},{3,2,1,0},{2,2,0,0}},{4125381956}CenterDot{{3,3,1,1},{3,2,1,0},{1,1,1,1},{1,0,1,0}},{4199610960}CenterDot{{3,3,2,2},{1,1,0,0},{3,3,2,2},{1,1,0,0}},{4293850596}CenterDot{{3,3,3,3},{3,2,3,2},{3,3,1,1},{3,2,1,0}}
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ResourceFunction["FindFiniteModels"][a(b·a)·(a·(b·c)),5,"Parallelize"->True]
Out[]=
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ResourceFunction["FindFiniteModels"][a(b·a)·(a·(b·c)),5,"Parallelize"->True]
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What is a common name for the relation a(b·a)·(a·(b·c))
The relation does not correspond to a widely recognized or standard name in conventional algebraic structures such as groups, rings, or fields. Instead, it appears to be a specific identity that might be part of a custom or specialized axiomatic system, potentially related to the Wolfram Axioms you are working with. Given the context of your work with and the usage of functions like , it seems you are exploring or defining a unique algebraic structure within the Wolfram Language framework. In such customized systems, relations like are typically foundational axioms or rules that define the behavior of the operations within that system. 1. Check Documentation or Source Material: If the Wolfram Axioms are part of a larger theory or framework, referring to the accompanying documentation or source material may provide specific names or interpretations for such relations.2. Define a Custom Name: If this relation is unique to your work, consider defining a custom name that reflects its role or behavior within your axiomatic system. For example, you might call it the Wolfram Associative-like Identity or another descriptive term.3. Explore Related Algebraic Structures: While this exact relation may not have a standard name, exploring similar identities in non-associative algebras (like Moufang algebras or alternative algebras) might provide useful insights or analogies.4. Consult the Wolfram Community: Engaging with the Wolfram Community or other forums may help you find others who have encountered similar relations or who can offer guidance based on the Wolfram Axioms. If you decide to name the relation for clarity in your work, you can define it as follows:
This approach helps in maintaining clarity, especially when collaborating with others or when revisiting the work in the future. If you have more specific questions about how this relation interacts with other axioms in your system or how to utilize it within Wolfram Language functions, feel free to ask!
What is a common name for the relation a(b·a)·(a·(b·c))
The relation you've provided: is an example of a Moufang identity. Moufang identities are a set of axioms that generalize the associative law found in groups. While groups strictly adhere to associativity, Moufang loops allow for certain non-associative operations while still maintaining some structure similar to groups. These identities ensure that even in the absence of full associativity, the system retains a level of manageable complexity and symmetry. There are several forms of Moufang identities, but some of the most common ones include:1. Left Moufang Identity: 2. Right Moufang Identity: 3. Alternative Moufang Identity: Your provided equation: mirrors the structure of Moufang identities by relating multiple products and incorporating both left and right multiplications. This specific form ensures a certain level of flexibility within the algebraic structure, similar to how Moufang loops operate. Moufang identities are significant in the study of non-associative algebraic structures, such as: Understanding and applying Moufang identities allows mathematicians and scientists to work with complex systems where traditional associative properties do not hold, providing a framework for exploring more generalized algebraic structures.In summary, the relation you've described is commonly known as a Moufang identity.Answer: It is one of the standard Moufang identities, commonly known as a Moufang identity.
• Moufang Loops: These are loops (a generalization of groups) that satisfy at least one of the Moufang identities.
• Alternative Algebras: These algebras satisfy the alternative laws, which are closely related to Moufang identities.
• Projective Geometry and Topology: Certain geometric and topological constructs utilize Moufang identities to maintain structural integrity without full associativity.