• • Steiner magma: Acommutative magma satisfyingx.xy =y.
    • Squag: anidempotent Steiner magma.^[1]
    • Sloop: a Steiner magma with distinguished element 1, such thatxx = 1.
  • Equivalential algebra: a magma satisfyingxx.y=y,xy.z.z=xy, andxy.xzz.xzz=xy.^[2]
  • Implicational calculus: a magma satisfyingxy.x=x,x.yz=y.xz, andxy.y=yx.x.^[3]
  • Equivalence algebra: anidempotentmagma satisfyingxy.x=x,x.yz=xy.xz, andxy.z.y.x =xz.y.x.^[4]
  • Semigroup: anassociative magma.
    • Equivalential calculus: a commutative semigroup satisfyingyyx=x.^[5]
      • Boolean group: a monoid withxx =identity element.
        • • Reductive group: analgebraic group such that theunipotent radical of the identity component ofS is trivial.
      • Logic algebra: a commutative monoid with aunary operation,complementation, denoted by enclosure in parentheses, and satisfyingx(1)=(1) and ((x))=x. 1 and (1) arelattice bounds forS.
        • MV-algebra: a logic algebra satisfying the axiom ((x)y)y = ((y)x)x.
        • Boundary algebra: a logic algebra satisfying (x)x=1 and (xy)y = (x)y, from which it can be proved that boundary algebra is adistributive lattice. (0)=1, (1)=0, ((x))=x andxx=x are now provable.
  • Order (algebra): anidempotent magma satisfyingyx=xy.x,xy=xy.y,x:xy.z=x.yz, andxy.z.y=xz.y. Hence idempotence holds in the following wide sense. For any subformulax of formulaz: (i) all but one instance ofx may be erased; (ii)x may be duplicated at will anywhere inz.
    • Band: an associative order algebra, and anidempotent semigroup.
      • Rectangular band: a band satisfying the axiomxyz =xz.
      • Normal band: a band satisfying the axiomxyzx =xzyx.

The following two structures form a bridge connectingmagmas andlattices:

• • Newman algebra: a ringoid that is also a shell. There is a unary operation, inverse, denoted by a postfix "'", such that x+x'=1 and xx'=0. The following are provable: inverse is unique,x"=x, addition commutes and associates, and multiplication commutes and is idempotent.
  • Semiring: a ringoid that is also a shell. Addition and multiplication associate, addition commutes.
    • Commutativesemiring: a semiring whose multiplication commutes.
  • Rng: a ringoid that is anAbelian group under addition and 0, and a semigroup under multiplication.
    • Ring: a rng that is a monoid under multiplication and 1.
      • Commutative ring: a ring with commutative multiplication.
        • Boolean ring: a commutative ring withidempotent multiplication,isomorphic toBoolean algebra.
      • Differential ring: A ring with an addedunary operation,derivation, denoted byprefix ∂ and satisfying theproduct rule, ∂(xy) = ∂xy+x∂y.
  • Bounded lattice: a lattice with two distinguished elements, thegreatest (1) and theleast element (0), such thatx∨1=1 andx∨0=x.
  • Involutive lattice: a lattice with a unary operation, denoted by postfix ', and satisfyingx"=x and (x∨y)' =x'∧y'.
  • Relatively complemented lattice:
  • Complemented lattice: a lattice with 0 and 1 such that for anyx there isy withx ∨y = 1 andx∧y = 0. Not definable by identities
  • Lattice with choice of complement: a lattice with a unary operation, [complementation]], denoted bypostfix ', such thatx∧x' = 0 and 1=0'. 0 and 1 boundS -- as well as the dual conditions.
    • Orthocomplemented lattice: a lattice with complementation satisfyingx" =x andx∨y=y ↔y'∨x'=x' (complementation isorder reversing).
      • Orthomodular lattice: an ortholattice such that (x ≤y) → (x ∨ (x^⊥ ∧y) =y) holds.
    • De Morgan algebra: a complemented lattice satisfyingx" =x and (x∨y)' =x'∧y'. Also a bounded involutive lattice.
  • Modular lattice: a lattice satisfying the modular identity,x∨(y∧(x∨z)) = (x∨y)∧(x∨z).
    • Metric lattice: not definable by identities
    • Projective lattice: not definable by identities
    • Arguesian lattice: a modular lattice satisfying the identity
    • Distributive lattice: a lattice in which each of meet and joindistributes over the other. Distributive lattices are modular, but the converse need not hold.
      • Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.
        • Boolean algebra with operators: a Boolean algebra with one or more added operations, usually unary. Let a postfix * denote any added unary operation. Then 0* = 0 and (x∨y)* =x*∨y*. More generally, all added operations (a) evaluate to 0 if any argument is 0, and (b) arejoin preserving, i.e., distribute over join.
          • Modal algebra: a Boolean algebra with a single added operator, themodal operator.
            • Derivative algebra: a modal algebra whose added unary operation, thederivative operator, satisfiesx**∨x*∨x =x*∨x.
            • Interior algebra: a modal algebra whose added unary operation, theinterior operator, satisfiesx*∨x =x andx** =x*. The dual is aclosure algebra.
              • Monadic Boolean algebra: a closure algebra whose added unary operation, theexistential quantifier, denoted by prefix ∃, satisfies the axiom ∃(∃x)' = (∃x)'. The dual operator, ∀x := (∃x')' is theuniversal quantifier.

Three structures whose intended interpretations arefirst order logic:

• Polyadic algebra: amonadic Boolean algebra with a second unary operation, denoted by prefixedS.I is anindex set,J,K⊂I. ∃ maps eachJ into thequantifier ∃(J).S mapsI→Itransformations into Booleanendomorphisms onS. σ, τ range over possible transformations; δ is theidentity transformation. The axioms are: ∃(∅)a=a, ∃(J∪K) = ∃(J)∃(K),S(δ)a =a,S(σ)S(τ) =S(στ),S(σ)∃(J) =S(τ)∃(J) (∀i∈I-J, such that σi=τi), and ∃(J)S(τ) =S(τ)∃(τ^-1J) (τinjective).^[6]
• Relation algebra:S, theCartesian square of some set, is a:

• Boolean algebra under join and complementation;
• Monoid under binarycomposition (infix •) and theidentity elementI such that 1=I '∨I;
• Residuated Boolean algebra by virtue of a second unary operation,converse (postfix^{${\displaystyle \stackrel{˘}{}}$}) and the axiom (A^{${\displaystyle \stackrel{˘}{}}$}•(A•B)')∨B ' =B '.

Converse is aninvolution and distributes over composition so that (A•B)^{${\displaystyle \stackrel{˘}{}}$} =B^{${\displaystyle \stackrel{˘}{}}$}•A^{${\displaystyle \stackrel{˘}{}}$}. Converse and composition eachdistribute over join.^[7]

• Cylindric algebra: Boolean algebra augmented by unary cylindrification operations.

Others:

• • • • • Coalgebra: thedual of a unital associative algebra.
        • Incidence algebra: an associative algebra such that the elements ofS are the functionsf [a,b]: [a,b]→R, where [a,b] is an arbitrary closed interval of a locally finiteposet. Vector multiplication is defined as aconvolution of functions.
        • Kac-Moody algebra: a Lie algebra, usually infinite-dimensional, definable bygenerators and relations through ageneralized Cartan matrix.
          • Generalized Kac-Moody algebra: a Kac-Moody algebra whosesimple roots may beimaginary.
          • Affine Lie algebra: a Kac-Moody algebra whosegeneralized Cartan matrix ispositive semi-definite and has corank 1.

These algebraic structures are not varieties, because the underlying set either has atopology or is amanifold, characteristics that are not algebraic in nature. This added structure must be compatible in some sense, however, with the algebraic structure. The case of when the added structure ispartial order is discussed above, under varieties.

Topology:

• Topological group: a group whoseS has atopology;
  • Discrete group: a topological group whose topology isdiscrete. Also a 0-dimensionalLie group.
• Topological vector space: anormed vector space whoseR has atopology.

Manifold:

• Lie group: a group whoseS has a smoothmanifold structure.

Let there be twoclasses:

• O whose elements areobjects, and
• M whose elements aremorphisms defined overO.

Letx andy be any two elements ofM. Then there exist:

• Twofunctions,c,d :M→O.d(x) is thedomain ofx, andc(x) is itscodomain.
• A binarypartial operation overM, calledcomposition and denoted byconcatenation.xy is definediffc(x)=d(y). Ifxy is defined,d(xy) =d(x) andc(xy) =c(y).

Category:Composition associates (if defined), andx hasleft andright identity elements, the domain and codomain ofx, respectively, so thatd(x)x =x =xc(x). Letting φ stand for one ofc ord, and γ stand for the other, then φ(γ(x)) = γ(x).IfO has but one element, the associated category is amonoid.

• Groupoid: Two equivalent definitions.
  • Category theory: Asmall category in which every morphism is anisomorphism. Equivalently, a category such that every elementx ofM,x(a,b), has an inversex(b,a); see diagram in section 2.2.
  • Algebraic definition: A group whose product is apartial function. Group product associates in that ifab andbc are both defined, thenab.c=a.bc. (a)a anda(a) are always defined. Also,ab.(b) =a, and (a).ab =b.

• Incidence algebra: Anassociative algebra, defined for any locally finiteposet andcommutative ring with unity. Part oforder theory.
• Group ring:
  • Group algebra:
• Path algebra: related to aquiver and adirected graph.
• Categorical algebra: anassociative algebra defined for any locally finite category andcommutative ring with unity. Generalizesgroup algebra andincidence algebra, as the concept ofcategory generalizesgroup andposet.

• Division ring (alsoskew field,sfield): aring such thatS-0 is agroup under multiplication.
  • Field: a division ring whose multiplication commutes. Recapitulating:S is an abelian group under addition and 0,S-0 is an abelian group under multiplication and 1≠0, and multiplicationdistributes over addition.x0 = 0 is a theorem.
    • Algebraically closed field: a field such that allpolynomial equations whose coefficients are elements ofS have allroots inS. This field is thecomplex numbers.
    • Ordered field: a field whoseS istotally ordered by '≤', so that (a≤b)→(a+c≤b+c) and (0≤a,b)→ (0≤ab).
      • Real closed field: an ordered real field such that for every elementx ofS, there exists ay such thatx =y² or -y². Allpolynomial equations of odd degree and whose coefficients are elements ofS, have at least oneroot that inS.
      • Real field: aDedekind complete ordered field.
        • Differential field: A real field with an addedunary operation,derivation, denoted by prefix ∂, distributing overaddition, ∂(x+y) = ∂x+ ∂y, and satisfying theproduct rule, ∂(xy) = ∂xy +x∂y.

• Part algebra: aBoolean algebra with no least element 0, so that the complement of 1 is not defined.

Two sets, Φ andD.

• Information algebra:D is a lattice, and Φ is a commutativemonoid under combination, anidempotent operation. The operation of focussing,f: ΦxD→Φ satisfies the axiomf(f(φ,x),y)=f(φ,x∧y) and distributes over combination. Every element of Φ has an identity element inD under focussing.

If the name of a structure in this section includes the word "arithmetic," the structure features one or both of thebinary operationsaddition andmultiplication. If both operations are included, the recursive identity defining multiplication usually links them. Arithmetics necessarily haveinfinitemodels.

• Cegielski arithmetic^[8]: A commutativecancellativemonoid under multiplication. 0 annihilates multiplication, andxy=1if and only ifx andy are both 1. Other axioms and one axiom schema govern order,exponentiation,divisibility, andprimality; consult Smorynski. Adding thesuccessor function and its axioms as per Dedekind algebra render addition recursively definable, resulting in a system with the expressive power ofRobinson arithmetic.

In the structures below, addition and multiplication, if present, arerecursively defined by means of aninjective operation calledsuccessor, denoted byprefix σ. 0 is the axiomaticidentity element for addition, and annihilates multiplication. Both axioms hold forsemirings.

• Dedekind algebra^[9], also called aPeano algebra: A pointed unary system by virtue of 0, the unique element ofS not included in therange ofsuccessor. Dedekind algebras are fragments of Skolem arithmetic.
  • Dedekind-Peano structure: A Dedekind algebra with anaxiom schema ofinduction.
    • Presburger arithmetic: A Dedekind-Peano structure withrecursiveaddition.

Arithmetics above this line aredecidable. Those below areincompletable.

• Robinson arithmetic: Presburger arithmetic withrecursivemultiplication.

• Peano arithmetic: Robinson arithmetic with anaxiom schema ofinduction. The semiring axioms forN (other thanx+0=x andx0=0, included in the recursive definitions of addition and multiplication) are now theorems.

• Heyting arithmetic: Peano arithmetic withintuitionist logic as the background logic.

• • Primitive recursive arithmetic: A Dedekind algebra with recursively defined addition, multiplication, exponentiation, and otherprimitive recursive operations as desired. A rule of induction replaces theaxiom of induction. The background logic lacksquantification and thus is notfirst-order logic.
  • Skolem arithmetic (Boolos and Jeffrey 2002: 73-76): Not an algebraic structure because there is no fixed set of operations of fixedadicity. Skolem arithmetic is a Dedekind algebra withprojection functions, indexed byn, whose arguments are functions and that return thenth argument of a function. Theidentity function is the projection function whose arguments are all unary operations. Composite operations of anyadicity, including addition and multiplication, may be constructed usingfunction composition andprimitive recursion.Mathematical induction becomes a theorem.
    • Kalmar arithmetic: Skolem arithmetic with different primitive functions.

The following arithmetics lack a connection between addition and multiplication. They are the simplest arithmetics capable of expressing allprimitive recursive functions.

• Baby Arithmetic^[10]: Because there is nouniversal quantification, there areaxiom schemes but no axioms. [n] denotesn consecutive applications ofsuccessor to 0. Addition and multiplication are defined by the schemes [n]+[p] = [n+p] and [n][p] = [np].
  • R^[11]: Baby arithmetic plus thebinary relations "=" and "≤". These relations are governed by the schemes [n]=[p] ↔n=p, (x≤[n])→(x=0)∨,...,∨(x=[n]), and (x≤[n])∨([n]≤x).

Nonvarieties cannot beaxiomatized solely withidentities andquasiidentities. Many nonidentities are of three very simple kinds:

1. The requirement thatS (orR orK) be a "nontrivial"ring, namely one such thatS≠{0}, 0 being the additiveidentity element. The nearest thing to an identity implyingS≠{0} is the nonidentity 0≠1, which requires that the additive and multiplicative identities be distinct.
2. Axioms involving multiplication, holding for all members ofS (orR orK) except 0. In order for an algebraic structure to be a variety, thedomain of each operation must be an entire underlying set; there can be nopartial operations.
3. "0 is not thesuccessor of anything," included in nearly all arithmetics.

Most of the classic results ofuniversal algebra do not hold for nonvarieties. For example, neither thefree field over any set nor thedirect product ofintegral domains exists. Nevertheless, nonvarieties often retain an undoubted algebraic flavor.

There are whole classes ofaxiomaticformal systems not included in this section, e.g.,logics,topological spaces, and this exclusion is in some sense arbitrary. Many of the nonvarieties below were included because of their intrinsic interest and importance, either by virtue of their foundational nature (Peano arithmetic), ubiquity (thereal field), or richness (e.g.,fields,normed vector spaces). Also, a great deal of theoretical physics can be recast using the nonvarieties calledmultilinear algebras.

The elements ofS arehigher order functions, and concatenation denotes the binary operation offunction composition.

• BCI algebra: a magma with distinguished element 0, satisfying the identities (xy.xz)zy = 0, (x.xy)y = 0,xx=0,xy=yx=0 →x=y, andx0 = 0 →x=0.
  • BCK algebra: a BCI algebra satisfying the identityx0 =x.x≤y, defined asxy=0, induces apartial order with 0 as least element.
• Combinatory logic: Acombinatorconcatenates upper case letters.Terms concatenate combinators and lower case letters. Concatenation is left and rightcancellative. '=' is anequivalence relation over terms. The axioms areSxyz =xz.yz andKxy =x; these implicitly define the primitive combinatorsS andK. The distinguished elementsI and1, defined asI=SK.K and1=S.KI, have the provable propertiesIx=x and1xy=xy. Combinatory logic has the expressive power ofset theory.^[12]
  • Extensional combinatory logic: Combinatory logic with the added quasiidentity (Wx=Vx)→(W=V), withW,V containing no instance ofx.

Three binary operations.

• Normed vector space: a vector space with anorm, namely a functionM→R that issymmetric,linear, andpositive definite.
  • Inner product space (alsoEuclidian vector space): a normed vector space such thatR is thereal field, whose norm is the square root of theinner product,M×M→R. Leti,j, andn be positive integers such that 1≤i,j≤n. ThenM has anorthonormal basis such thate_i•e_j = 1 ifi=j and 0 otherwise. Seefree module.
  • Unitary space: Differs from inner product spaces in thatR is thecomplex field, and the inner product has a different name, thehermitian inner product, with different properties:conjugate symmetric,bilinear, andpositive definite.^[13]
• Graded vector space: a vector space such that the members ofM have adirect sum decomposition. Seegraded algebra below.

• • • Group Hopf algebra:
• Graded algebra: an associative algebra withunital outer product. The members ofV have adirectram decomposition resulting in their having a "degree," with vectors having degree 1. Ifu andv have degreei andj, respectively, the outer product ofu andv is of degreei+j.V also has a distinguished member0 for each possible degree. Hence all members ofV having the same degree form anAbelian group under addition.
  • Tensor algebra: A graded algebra such thatV includes all finite iterations of a binary operation overV, called thetensor product. All multilinear algebras can be seen as special cases of tensor algebra.
    • Exterior algebra (alsoGrassmann algebra): a graded algebra whoseanticommutative outer product, denoted by infix ∧, is called theexterior product.V has anorthonormal basis.v₁ ∧v₂ ∧ ... ∧v_k = 0 if and only ifv₁, ...,v_k arelinearly dependent. Multivectors also have aninner product.
      • Clifford algebra: an exterior algebra with a symmetricbilinear formQ:V×V→K. The special caseQ=0 yields an exterior algebra. The exterior product is written 〈u,v〉. Usually, 〈e_i,e_i〉 = -1 (usually) or 1 (otherwise).
      • Geometric algebra: an exterior algebra whose exterior (calledgeometric) product is denoted by concatenation. The geometric product of parallel multivectors commutes, that of orthogonal vectors anticommutes. The product of a scalar with a multivector commutes.vv yields a scalar.
        • Grassmann-Cayley algebra: a geometric algebra without an inner product.