Two first-order structures are elementarily equivalent when they satisfy the same first-order sentences. They need not be isomorphic or embed into one another. Every elementary embedding implies elementary equivalence of its source and target.

Given nonempty first-order structures ${\mathcal{M}}_i$ and a proper filter on a set $F$, a reduced product identifies product functions $f,g$ when $\{i:f(i)=g(i)\}\in F$. Operations are interpreted coordinatewise, and relations hold when their coordinate truth sets belong to $F$. For an ultrafilter this is an ultraproduct. Primitive positive formulas transfer through an arbitrary proper filter, whereas logical disjunction and logical negation can fail to transfer.

For a first-order theory $T$ and $A\models T_{\forall }$, an algebraically prime extension is a structure embedding $i:A\to P\models T$ such that every embedding $j:A\to N\models T$ factors as $j=h\circ i$ for some structure embedding $h:P\to N$. The embeddings need not be elementary or unique. A theory has algebraically prime models when every such $A$ has an extension of this kind.

This first-order theory has as models the nonzero torsion-free divisible Abelian groups. Rational division equips them with the structure of vector spaces over the rational numbers. A rational divisible hull gives an algebraically prime extension of each nonzero torsion-free base, while the zero base uses a one-dimensional rational space.

The theory of formally real fields consists of the field axioms and all sentences asserting that $-1$ is not a sum of $m$ squares, one for each positive integer $m$. The real closure theorem and model completeness make the Theory of real closed fields its model companion.

In each round the Spoiler selects an element from either structure and the Duplicator responds in the other. The paired selected elements must preserve all atomic information and equality, giving a partial isomorphism of structures. In the infinite game the Duplicator wins by avoiding failure at every finite stage. In a timed game the Spoiler first announces the number of rounds, allowing a different response strategy for each horizon.

A finitary algebraic theory specifies sorts, finite-arity operations and equations between terms. Its internal models can be interpreted in any category with finite products. Finitely presented set-based models generate arbitrary models under filtered colimits and determine its classifying topos.

A recursive presentation identifies the domain of a countable structure with a computable set, with its basic operations given by total computable functions and its basic relations decidable. In a fixed finite signature this makes evaluation of atomic formulas effective. A countable structure may have a recursive presentation of a structure only for some choices of presentation, or none at all.

The mechanism assigns probabilities to observation patterns $R$ conditional on complete measurements $D$: $p(R\mid D)$. Whether it depends on missing values distinguishes missing completely at random, missing at random and missing not at random.

With canonical tree-level global supersymmetry, neutral F-term breaking and no D-term mass shifts, the tree-level supertrace mass sum rule applies in conserved electric and color charge blocks. The corresponding scalar squared-mass average equals the fermion squared-mass average, so not every scalar partner can lie above that average. This obstructs making every squark heavy through direct canonical visible-sector breaking while retaining light quarks. Effective soft supersymmetry breaking, noncanonical interactions, radiative effects or supergravity invalidate the assumptions of that argument.

In the MSSM, each fermion generation has chiral superfields $Q:(3,2,1/6)$, $U^c:(\bar{3},1,-2/3)$, $D^c:(\bar{3},1,1/3)$, $L:(1,2,-1/2)$, and $E^c:(1,1,1)$. This uses hypercharge normalized by $Q_{\mathrm{electric}}=T^3+Y$ and only left-handed Weyl spinors. Two Higgs chiral doublets have charges $\pm 1/2$. The three vector superfields transform as $(8,1,0)$, $(1,3,0)$ and $(1,1,0)$, supplying the gauge bosons and gauginos.

A Higgs chiral doublet is a weak-doublet chiral superfield containing a Higgs complex scalar field, a higgsino and an auxiliary field. In the MSSM, $H_u$ has hypercharge $+1/2$ and $H_d$ has hypercharge $-1/2$. Their paired charges enable higgsino anomaly cancellation and the holomorphic need for two Higgs chiral doublets.

A single weak-doublet higgsino of hypercharge $+1/2$ contributes $1/4$ to the cubic hypercharge gauge anomaly, $1/4$ to the weak-squared hypercharge coefficient with fundamental index $1/2$, and $1$ to the mixed gauge-gravitational anomaly. A second Higgs chiral doublet of opposite hypercharge cancels all three through its higgsino. The pair also contributes an even number of weak doublets, avoiding a new Witten SU(2) anomaly. Higgs complex scalar fields do not cancel chiral fermion gauge anomalies.

A renormalizable MSSM superpotential is a holomorphic function of chiral superfields and cannot use their conjugates. Gauge-invariant up-type Yukawa couplings require $H_u$ of hypercharge $+1/2$, while down-type and charged-lepton Yukawa couplings require $H_d$ of hypercharge $-1/2$. A single doublet and its conjugate cannot play both roles in the superpotential. This is distinct from higgsino anomaly cancellation.

A Lorentzian metric is a smooth nondegenerate symmetric metric tensor with one negative and all other positive directions, or its opposite-sign convention. At each point, a nonzero tangent is timelike, null or spacelike according as $g(v,v)$ is negative, zero or positive in the first convention. Unlike a Riemannian metric, it gives a light cone rather than a positive norm. Minkowski spacetime is its constant flat model, and proper time along a timelike curve is $\int \sqrt{-g(\dot{\gamma },\dot{\gamma })}du$.

For $0<\alpha <\beta$, define $-\beta \le \mu \le -\alpha \le \lambda$ by $R^2/(\tau +\alpha )+z^2/(\tau +\beta )=1$. Then $R^2=(\lambda +\alpha )(\mu +\alpha )/(\alpha -\beta )$ and $z^2=(\lambda +\beta )(\mu +\beta )/(\beta -\alpha )$. Constant $\lambda$ surfaces are confocal prolate spheroids and constant interior $\mu$ surfaces are two-sheeted hyperboloids, with foci $z=\pm \sqrt{\beta -\alpha }$. The coordinates are orthogonal away from their degeneracies. The squared scale factors are $h_{\lambda }^2=(\lambda -\mu )/[4(\lambda +\alpha )(\lambda +\beta )]$, $h_{\mu }^2=(\mu -\lambda )/[4(\mu +\alpha )(\mu +\beta )]$ and $h_{\varphi }^2=R^2$. The two signs of $z$ are coordinate branches, joined at the equatorial fold $\mu =-\beta$.

For a nondegenerate metric tensor, differentiating $g_{ac}{g}^{cb}={\delta }_a{}^b$ gives $\delta g_{ab}=-g_{ac}{g}_{bd}\delta g^{cd}$. Combining this with the Jacobi determinant derivative formula proves the displayed volume-density variation. It is the determinant contribution when defining a stress-energy tensor by varying the inverse metric tensor.

A meson with zero spin angular momentum and negative parity. The pion and kaon are familiar examples. Between two pseudoscalar mesons, parity conservation in the strong interaction forces the axial current matrix element to vanish, while the vector current has pseudoscalar-to-pseudoscalar form factors. The weak decay as a whole need not conserve parity.

The common zero locus of constraint functions $C_a$ in phase space is the constraint surface. A physical trajectory lies on this surface, but off-surface identities are often needed when analyzing a constraint algebra.

Second-order arithmetic has variables for natural numbers and for sets of natural numbers. Subsystems restrict comprehension and induction; arithmetical transfinite recursion theory is one important subsystem stronger than the first-order system Peano arithmetic.