[Paper Review] Implication Zroupoids and Identities of Associative Type

Juan M. Cornejo, Hanamantagouda P. Sankappanavar|arXiv (Cornell University)|Oct 29, 2017

Advanced Topics in Algebra7 references3 citations

TL;DR

This paper provides a complete classification of subvarieties of implication zroupoids (I-zroupoids) defined by identities of associative type of length 3, proving there are exactly 8 distinct such subvarieties. It further shows that among symmetric I-zroupoids, exactly 3 distinct subvarieties arise from such identities, forming a strict inclusion chain: Boolean algebras ⊂ semilattices with least element ⊂ symmetric I-zroupoids.

ABSTRACT

An algebra $\mathbf A = \langle A, o, 0 angle$, where $ o$ is binary and $0$ is a constant, is called an implication zroupoid ($\mathcal I$-zroupoid, for short) if $\mathbf A$ satisfies the identities: $(x o y) o z \approx [(z' o x) o (y o z)']'$ and $ 0'' \approx 0$, where $x' : = x o 0$, and $\mathcal I$ denotes the variety of all $\mathcal I$-zroupoids. An $\mathcal I$-zroupoid is symmetric if it satisfies $x'' \approx x$ and $(x o y')' \approx (y o x')'$. The variety of symmetric $\mathcal I$-zroupoids is denoted by $\mathcal S$. An identity $p \approx q$, in the groupoid language $\langle o angle$, is called an identity of associative type of length $3$ if $p$ and $q$ have exactly 3 (distinct) variables, say x,y,z, and are grouped according to one of the two ways of grouping: (1) $\star o (\star o \star)$ and (2) $(\star o \star) o \star$, where $\star$ is a place holder for a variable. A subvariety of $\mathcal I$ is said to be of associative type of length $3$, if it is defined, relative to $\mathcal I$, by a single identity of associative type of length $3$. In this paper we give a complete analysis of the mutual relationships of all subvarieties of $\mathcal I$ of associative type of length $3$. We prove, in our main theorem, that there are exactly 8 such subvarieties of $\mathcal I$ that are distinct from each other and describe explicitly the poset formed by them under inclusion. As an application of the main theorem, we derive that there are three distinct subvarieties of the variety $\mathcal S$, each defined, relative to $\mathcal S$, by a single identity of associative type of length $3$.

Motivation & Objective

• To fully characterize all subvarieties of the variety of implication zroupoids (I) that are defined by a single identity of associative type of length 3.
• To determine the inclusion relationships among these subvarieties and establish their mutual structure.
• To analyze the restriction of these identities to the subvariety of symmetric I-zroupoids (S), identifying distinct subvarieties within S.
• To clarify the role of identities such as (A1), (A11), and (A14) in defining these subvarieties and their logical relationships.
• To resolve open questions about the lattice of subvarieties of I by providing a complete and explicit poset structure for this specific class of identities.

Proposed method

• Define implication zroupoids (I-zroupoids) as algebras ⟨A, →, 0⟩ satisfying two identities: (I) (x→y)→z ≈ [(z′→x)→(y→z)′]′ and (I0) 0′′ ≈ 0, where x′ := x→0.
• Introduce the concept of identities of associative type of length 3: identities with exactly three distinct variables grouped as (x→y)→z or x→(y→z), forming 14 possible identities.
• Use equational reasoning and lattice-theoretic techniques to analyze inclusion relations between subvarieties defined by these identities.
• Apply the main theorem (Theorem 4.2) to classify all 14 possible identities of associative type of length 3, proving they generate exactly 8 distinct subvarieties of I.
• Restrict the analysis to symmetric I-zroupoids (S), defined by x′′ ≈ x and (x→y′)′ ≈ (y→x′)′, and derive the subvarieties of S induced by the same identities.
• Use model-theoretic arguments and counterexamples (e.g., 4-element algebras) to verify strict inclusions and non-inclusions, such as BA ⊈ S₁₄.

Experimental results

Research questions

• RQ1 How many distinct subvarieties of implication zroupoids are defined by a single identity of associative type of length 3?
• RQ2 What is the complete inclusion structure (poset) of these subvarieties within the variety of implication zroupoids?
• RQ3 Which of these subvarieties are distinct when restricted to the subvariety of symmetric implication zroupoids?
• RQ4 What is the relationship between the variety of Boolean algebras and the subvarieties of symmetric implication zroupoids defined by associative-type identities?
• RQ5 Are there identities of associative type that define the same subvariety in both I and S, and if so, how do they relate?

Key findings

• There are exactly 8 distinct subvarieties of the variety of implication zroupoids (I) that are defined by a single identity of associative type of length 3.
• The poset of these 8 subvarieties forms a complete inclusion hierarchy, with the minimal element being the variety of semilattices with least element (SL) and the maximal being the full variety I.
• When restricted to symmetric I-zroupoids (S), only 3 distinct subvarieties arise from identities of associative type of length 3: SL, S₁₄, and S.
• These three subvarieties satisfy the strict inclusion chain: SL ⊂ S₁₄ ⊂ S, with S₁₄ being a proper intermediate variety.
• The variety of Boolean algebras (BA) is not contained in S₁₄, as shown by a counterexample algebra on 4 elements.
• The subvariety S₁₄ is defined by the identity (A14): (x→y)→(z→x) ≈ (x→z)→(y→x), and is strictly between SL and S.