Free-Forgetful Adjunction in Effect Algebras

• Free-forgetful adjunction is an adjoint pair where the left functor freely adjoins a top element and orthosupplementation, while the right functor forgets this extra structure.
• The unitization functor constructs effect algebras from generalized effect algebras by extending the structure with a canonical greatest element and orthosupplementation.
• The adjunction is monadic, meaning effect algebras are precisely the Eilenberg-Moore algebras for the induced monad, grounding the construction in categorical logic.

A free-forgetful adjunction is an adjoint pair of functors in which the right adjoint “forgets” a designated part of the algebraic or categorical structure, while the left adjoint “freely adds” that structure in a canonical way. In the context of the categories of generalized effect algebras (GEffAlg) and effect algebras (EffAlg), this adjunction is realized by the unitization construction, which freely adjoins a top element and orthosupplementation to a generalized effect algebra, transforming it into an effect algebra. The forgetful functor simply discards this extra structure. The adjunction is monadic, so the category of effect algebras can be reconstructed as the category of algebras for the monad induced by this adjunction (Jenča, 2017).

1. Categories: Generalized Effect Algebras and Effect Algebras

A generalized effect algebra (GEffAlg) is a partial algebra $P = (P;0,\oplus)$ where $0 \in P$ is a constant and $\oplus$ is a binary partial operation defined on a domain ("orthogonality" $\perp$), with the following axioms:

• (P1) $a \perp b \implies b \perp a$ and $a \oplus b = b \oplus a$,
• (P2) $b \perp c$ and $a \perp (b \oplus c) \implies a \perp b$; $(a \oplus b) \perp c$ and $(a \oplus b) \oplus c = a \oplus (b \oplus c)$,
• (P3) $a \perp 0$ and $a \oplus 0 = a$,
• (P4) $a \oplus b = a \oplus c \implies b = c$,
• (P5) $a \oplus b = 0 \implies a = 0$.

The canonical partial ordering is $a \leq b$ iff $\exists c: a \oplus c = b$. Morphisms are maps preserving $0$ and partial sums: $f(0) = 0$, and $a \perp b$ implies $f(a) \perp f(b)$ and $f(a \oplus b) = f(a) \oplus f(b)$.

An effect algebra (EffAlg) is a partial algebra $E = (E;0,1,\oplus)$ such that $(E;0,\oplus)$ is a generalized effect algebra, and $1 \in E$ is a greatest element ($x \leq 1$ for all $x$). Morphisms $\varphi: E_1 \rightarrow E_2$ are GEffAlg-morphisms preserving $1$.

2. The Forgetful Functor

The forgetful functor $U: \text{EffAlg} \rightarrow \text{GEffAlg}$ operates as follows:

• Objects: $U(E;0,1,\oplus) = (E;0,\oplus)$, forgetting the top element $1$;
• Morphisms: $U(\varphi) = \varphi$.

$U$ strictly forgets the nullary operation $1$ and is faithful.

3. The Unitization Functor

The left adjoint $F: \text{GEffAlg} \rightarrow \text{EffAlg}$—the unitization functor—constructs the effect algebra $F(P)$ from a generalized effect algebra $P = (P;0,\oplus)$ as follows:

• Underlying set: $F(P) = P \sqcup P^*$, where $P^* = \{ a^* \mid a \in P \}$ is a disjoint copy.
• Distinguished constants: $0 \in P \subset F(P)$, and $1 := 0^* \in P^*$.
• Partial sum:
  • $a \perp_F b$ iff $a \perp_P b$, $a \oplus_F b := a \oplus_P b \in P$.
  • $a \perp_F b^*$ iff $a \leq b$; $a \oplus_F b^* := (b \ominus_P a)^*$.
  • $a^* \perp_F b$ iff $a \geq b$; $a^* \oplus_F b := (a \ominus_P b)^*$.
  • $a^* \perp_F b^*$ is never defined.

Functoriality: for $f: P \to Q$, $F(f)(a) = f(a)$, $F(f)(a^*) = f(a)^*$.

4. The Adjunction and Its Universal Property

The adjunction $F \dashv U$ is realized via the unit and counit natural transformations:

• Unit $\eta: \text{Id}_{\text{GEffAlg}} \rightarrow U F$: for $P$, $\eta_P(x) = x$ (the inclusion).
• Counit $$\varepsilon: F U \rightarrow \text{Id}_{\text{EffAlg}}$$: for $E$, $\varepsilon_E(x) = x$, $\varepsilon_E(x^*) = x'$ (orthosupplement in $E$).

There is a natural bijection:

$$\text{EffAlg}(F(P), E) \cong \text{GEffAlg}(P, U(E))$$

given by restricting an EffAlg-morphism $\varphi: F(P) \to E$ to $P$ (via the unit), and conversely by extending $\psi: P \to U(E)$ to $\chi: F(P) \to E$ defined by $\chi(a) = \psi(a)$, $\chi(a^*) = (\psi(a))'$. These assignments are mutual inverses.

The triangle identities

$$\varepsilon_{F(P)} \circ F(\eta_P) = \text{id}_{F(P)}, \quad U(\varepsilon_E) \circ \eta_{U(E)} = \text{id}_{U(E)}$$

hold by inspection on generators.

5. Monadicity and Beck's Theorem

An adjunction $F \dashv U$ is monadic if $U$ creates the coequalizers of $U$-split pairs and reflects isomorphisms (Beck's monadicity theorem). In this setting,

• $U: \text{EffAlg} \to \text{GEffAlg}$ creates all coequalizers: if $h: B \to Z$ is a coequalizer in GEffAlg, then $Z$ inherits a top element $h(1_B)$ making it an effect algebra, and $h$ preserves it.
• $U$ reflects isomorphisms: $U$ is faithful and strictly forgets only $1$.

Therefore, by Beck’s theorem, the adjunction is monadic.

6. The Induced Monad and Eilenberg-Moore Category

The induced monad $T = U F$ on $\text{GEffAlg}$ is given by:

• $T(P) = U(F(P)) = P \sqcup P^*$, with unit $\eta_P(x) = x$ and multiplication $\mu_P = U(\varepsilon_{F(P)})$ defined by:
• $\mu_P(a) = a$,
• $\mu_P(a^*) = a^*$,
• $\mu_P(a^{**}) = a$,
• $\mu_P((a^*)^*) = a^*$.

A $T$-algebra is a pair $(P, \alpha)$ where $\alpha: T(P) \to P$ satisfies the usual monad unit and associativity axioms. Giving such an $\alpha$ is equivalent to endowing $P$ with a top element $1$ and an orthosupplement operation $x \mapsto x'$ satisfying the axioms for effect algebras. The category of $T$-algebras is isomorphic to EffAlg, and $U$ is (up to equivalence) the forgetful functor from EffAlg to GEffAlg. Thus,

$\text{GEffAlg}^T \simeq \text{EffAlg}$

and $F \dashv U$ is monadic.

7. Significance and Connections

The free-forgetful adjunction between generalized effect algebras and effect algebras encapsulates the process of freely adjoining a top element and the associated orthosupplementation in quantum structures and effect-algebraic formulations of logic. The explicit unitization construction due to Hedl, as formalized in this adjunction, is canonical and monadic, meaning that effect algebras are precisely the Eilenberg-Moore algebras for the monad induced by the unitization process. This provides the categorical foundation for constructions and studies involving extensions, completions, and presentations of effect-theoretic objects (Jenča, 2017).